We experimentally demonstrate the visualization of exciton energy transport in a non-uniformly strained WSe2 monolayer by monitoring the propagation of exciton densities via spectrally, temporally, and spatially resolved photoluminescence measurements at room temperature. Our measurements indicate that excitons in the WSe2 monolayer exhibit anomalous diffusion due to disorder in the system, which leads to a time-varying diffusion coefficient. In addition, we show that the sensitivity of monolayer WSe2's bandgap to strain gives rise to a built-in excitonic potential that results in the funneling of excitons towards high tensile strain points. The observed drift and diffusion agree reasonably with our proposed model that takes into account the strain field on the monolayer to describe the exciton dynamics.

Transition metal dichalcogenides (TMDs) have received special attention in the last decade due to their potential to enable optoelectronic devices with novel functionalities.1–3 In addition, the ability to tune their band structures via strain, temperature, and pressure along with their room-temperature stable excitonic resonances has made these materials outstanding candidates for excitonic device applications.4,5 Various devices exploiting TMD's optoelectronic potential have been demonstrated in recent years.1,6–14 Typically, these devices employ an electrical bias to dissociate the excitons and drive the charges towards a contact. In contrast, purely excitonic devices leverage TMD's large binding energy15–18 and highly tunable band structure19,20 to control the transport of exciton densities at room-temperature without dissociating them to collect the charges. Hence, it is critical that the transport properties of excitons in monolayer TMDs be well characterized to enable the design of next-generation excitonic devices.

The transport of excitonic energy under strain fields in TMDs is particularly interesting due to the large tunability of their bandgap via strain and the resulting capability to manipulate their motion via spatially modulated strain fields. Several studies exploring the effect of in-plane strain on monolayer TMDs' band structures have shown that for small strain levels, their direct bandgap varies linearly with strain.21–24 Their large sensitivity to strain can enable the design of excitonic potentials via the spatial modulation of their bandgap with non-uniform strain. Such modulation can result in the “funneling” of excitonic energy in the direction of the strain gradient.25 While there have been attempts to characterize the transport of excitons in bulk and monolayer TMDs,26–30 their combined drift and diffusion under strain gradients have not been studied in these materials. Here, we simultaneously quantify the diffusion and drift of excitons under the strain gradients created by transferring a WSe2 monolayer over a nanostructured substrate.

Monolayer WSe2 samples were prepared via mechanical exfoliation31 and were transferred over SiO2 nano-pillars on a Si substrate. The resulting strain profiles on the WSe2 monolayers were obtained by measuring the local PL spectrum across an area strained by a SiO2 pillar. Figure 1(a) shows an optical image of a WSe2 monolayer over an ∼1.5 μm diameter, ∼250 nm tall SiO2 pillar, and Fig. 1(b) shows the monolayer's integrated PL spectra from the area enclosed by the square on the optical image. The PL spectra were integrated between 622 nm and 881 nm since the peak intensity of WSe2 occurs typically at around 750 nm. The integrated spectra were normalized with respect to their maximum value within the enclosed area to emphasize the effect of the pillar.

FIG. 1.

(a) Optical image of a WSe2 monolayer transferred over an ∼1.5 μ m diameter, ∼250 nm tall SiO2 pillar on a Si substrate. (b) Normalized integrated PL spectra of the WSe2 monolayer strained by the SiO2 pillar. The monolayer's scanned area is enclosed by the square shown in (a). The local PL spectra were integrated between 622 nm and 881 nm and they were normalized with respect to the maximum integrated PL value within the enclosed area. (c) Bandgap energy shift/strain map of the monolayer. The white dot on the strain map indicates the location of the baseline spectrum where the monolayer is assumed to be unstrained. The strain sensitivity to bi-axial strain of the WSe2 monolayer's bandgap was obtained from Ref. 24. (d) Local PL spectra's centroid map. The excitation laser used for mapping the strain field was a 532 nm, CW diode laser.

FIG. 1.

(a) Optical image of a WSe2 monolayer transferred over an ∼1.5 μ m diameter, ∼250 nm tall SiO2 pillar on a Si substrate. (b) Normalized integrated PL spectra of the WSe2 monolayer strained by the SiO2 pillar. The monolayer's scanned area is enclosed by the square shown in (a). The local PL spectra were integrated between 622 nm and 881 nm and they were normalized with respect to the maximum integrated PL value within the enclosed area. (c) Bandgap energy shift/strain map of the monolayer. The white dot on the strain map indicates the location of the baseline spectrum where the monolayer is assumed to be unstrained. The strain sensitivity to bi-axial strain of the WSe2 monolayer's bandgap was obtained from Ref. 24. (d) Local PL spectra's centroid map. The excitation laser used for mapping the strain field was a 532 nm, CW diode laser.

Close modal

The direct bandgap of monolayer TMDs is expected to decrease (increase) linearly with tensile (compressive) strain24 which should be evidenced by red (blue) shifts in their PL spectra. Moreover, an enhancement of their PL intensity at high tensile strain points is expected due to the funneling of excitons towards the lower bandgap energy (red-shifted) regions of the monolayer. Such an enhancement in the WSe2 monolayer's local PL intensity is observed in Fig. 1(b) around the SiO2 pillar, which is consistent with the funneling of excitons towards the points of the highest tensile strain. The additional high strain points on the strain map shown in Fig. 1(c) could be the result of substrate roughness32 or polymer residue from the fabrication process. AFM images of the sample can be found in the supplementary material showing the possible origin of such high-strain points.

The strain field on the WSe2 monolayer was estimated by calculating the shift of the local PL spectra's centroids with respect to an unstrained point on the monolayer. The spectral shifts were converted to bandgap energy shifts and strain using the sensitivity of WSe2's direct bandgap to strain (−55 meV/%) from Ref. 24. Figure 1(c) shows the WSe2 monolayer's bandgap shift as a result of the applied strain. The white dot indicates the location of the unstrained PL spectrum baseline. In the supplementary material, we compare the strain calculated from PL spectra's centroid and peak shifts and show that the strain estimates are nearly identical.

The transport of excitons on an unstrained point of the WSe2 monolayer was visualized first to set a reference for the transport of excitons on the strained portion of the monolayer. A map of the time-dependent exciton density as a function of position was constructed using the technique from Refs. 33 and 34. The spatial information is obtained by scanning a detector across the PL emission spot resulting from a pulsed excitation while the temporal information is obtained via time-correlated single photon counting. Figure 2(b) shows the exciton density as a function of time and position where the excitation was fixed at an unstrained point as indicated by the cross in Fig. 2(a). The normalized exciton density was fitted with Gaussians to extract the time-dependent mean-squared displacement (MSD) of the exciton density.

FIG. 2.

(a) Local PL centroid map showing the point of excitation on the WSe2 monolayer indicated by the plus (+) sign. (b) Normalized exciton density as a function of position and time obtained with an avalanche photodiode detector (APD) and time-correlated single photon counting (TCSPC). (c) Time slices of the normalized exciton density fitted with Gaussians showing the broadening of the density; σ(t) denotes the time-dependent Gaussian half width. (d) Mean squared displacement (MSD) of the exciton density as a function of time showing sub-diffusive transport on the unstrained WSe2 monolayer, which is evidenced by the anomalous coefficient α<1, and resulting in the time-varying diffusion coefficient shown in the inset. The MSD is defined as Δx2t=σ2tσ2(0). The error bars represent the fitting error when fitting the measured exciton density with Gaussians.

FIG. 2.

(a) Local PL centroid map showing the point of excitation on the WSe2 monolayer indicated by the plus (+) sign. (b) Normalized exciton density as a function of position and time obtained with an avalanche photodiode detector (APD) and time-correlated single photon counting (TCSPC). (c) Time slices of the normalized exciton density fitted with Gaussians showing the broadening of the density; σ(t) denotes the time-dependent Gaussian half width. (d) Mean squared displacement (MSD) of the exciton density as a function of time showing sub-diffusive transport on the unstrained WSe2 monolayer, which is evidenced by the anomalous coefficient α<1, and resulting in the time-varying diffusion coefficient shown in the inset. The MSD is defined as Δx2t=σ2tσ2(0). The error bars represent the fitting error when fitting the measured exciton density with Gaussians.

Close modal

In systems where excitons diffuse freely, the MSD denoted by Δx2t is proportional to time according to Δx2tσ2tσ2(0)=2Dt, where 2σ(t) represents the width of the Gaussian exciton density, D is the diffusion coefficient, and t denotes the time.35,36 However, when excitons diffuse in a disordered environment, the MSD does not increase linearly with time, and the diffusion coefficient becomes time-varying. In such cases, the MSD is better described by the power law model Δx2t=Γtα, where Γ and α are known as the transport factor and anomalous coefficient, respectively.35–39 Accordingly, the time-dependent diffusion coefficient is defined as Dt=1/2Δx2t/t=αΓtα1/2.35 

In addition to the typical defects found in semiconductors, the two-dimensional nature of monolayer TMDs makes them especially susceptible to defects such as folding and wrinkling.40 This means that the type of substrate supporting the monolayer can radically affect the amount of disorder experienced by excitons and hence their diffusion.41,42 As a result, a sub-diffusive regime characterized by an anomalous coefficient α<1 and a diffusion coefficient varying between 1.8 and 0.6 cm2/s is identified on the unstrained WSe2 monolayer as shown in Fig. 2(d). In addition, the peak's position of the exciton density remains virtually unchanged in space, which is consistent with the absence of strain gradients that are expected to generate a drift of the density's peak.

On the other hand, when the excitation was positioned at a strained point on the monolayer near the SiO2 pillar, the exciton density's peak did not remain localized. Specifically, when the excitation was positioned just above the high tensile strain point on top of the pillar and the PL beam was monitored from top to bottom, the exciton density's peak shifted down and towards the high tensile strain point on the monolayer as shown in Fig. 3(b). Similarly, when the excitation was placed just to the left of the high tensile strain point on top of the pillar and the PL beam was monitored from left to right, the exciton density's peak shifted to the right and towards the high tensile strain point as shown in Fig. 3(c). A shift in the exciton density's peak position is expected to occur due to the strain gradient on the monolayer. Figure 3(d) shows the PL spectra measured at three positions along the x-axis near the SiO2 pillar.

FIG. 3.

(a) PL centroid map of the WSe2 monolayer zoomed in around the SiO2 pillar and showing the excitation positions indicated by the plus (+) signs. (b) Normalized exciton density as a function of position and time when the excitation was placed above the pillar and the APD was scanned along the vertical direction from top to bottom. (c) Normalized exciton density as a function of position and time when the excitation was placed to the left of the pillar and the APD was scanned along the horizontal direction from left to right. (d) PL spectra of the WSe2 monolayer measured at the positions labeled by X1 (blue), X2 (red), and X3 (green) in (a). The dashed vertical lines indicate the peak wavelengths of each PL spectrum. The tensile strain caused by the SiO2 pillar is evidenced by the red-shifts and the enhancement of the PL peak intensity as the strain increases. The PL spectra were fit with two Gaussians as WSe2 typically shows exciton and trion resonances. In the supplementary material, we show the details of these fits to demonstrate that the effect of doping on the observed spectral shifts is minimal. The excitation used for these measurements was a 405 nm pulsed laser.

FIG. 3.

(a) PL centroid map of the WSe2 monolayer zoomed in around the SiO2 pillar and showing the excitation positions indicated by the plus (+) signs. (b) Normalized exciton density as a function of position and time when the excitation was placed above the pillar and the APD was scanned along the vertical direction from top to bottom. (c) Normalized exciton density as a function of position and time when the excitation was placed to the left of the pillar and the APD was scanned along the horizontal direction from left to right. (d) PL spectra of the WSe2 monolayer measured at the positions labeled by X1 (blue), X2 (red), and X3 (green) in (a). The dashed vertical lines indicate the peak wavelengths of each PL spectrum. The tensile strain caused by the SiO2 pillar is evidenced by the red-shifts and the enhancement of the PL peak intensity as the strain increases. The PL spectra were fit with two Gaussians as WSe2 typically shows exciton and trion resonances. In the supplementary material, we show the details of these fits to demonstrate that the effect of doping on the observed spectral shifts is minimal. The excitation used for these measurements was a 405 nm pulsed laser.

Close modal

To explain our observations, we developed a model that encompasses the diffusion, drift, and relaxation of the exciton density of the WSe2 monolayer based on Boltzmann's transport theory with the relaxation time approximation and assuming a uniform exciton mobility.43–45 The resulting differential equation describing the exciton dynamics is given by

tnx,t=nx,tτ+Dt2x2nx,t+μϵxnx,txϵx,
(1)

where the first, second, and third terms to the right-hand side of the equal sign represent the relaxation of the exciton density denoted by nx,t with relaxation time τ, the diffusion of the exciton density with time-varying diffusion coefficient D(t), and the drift of the exciton density with strain mobility μϵ due to the strain gradient ϵx/x,respectively. The strain mobility is simply defined as μϵμEg/ϵ=|vdϵx/x1|, where μ is the traditional mobility, Eg/ϵ represents the sensitivity of the monolayer's direct bandgap to strain, and vd denotes the density's drift velocity due to the applied strain gradient. The units of the strain mobility are length squared per unit time and percent strain, typically expressed in cm2 s−1%–1. The traditional mobility is defined as μt¯/m*, where t¯ is the mean free time of the excitons, and m* is their effective mass.46 

In our case, the strain mobility is not uniform throughout the WSe2 monolayer because the non-uniform strain field modifies the monolayer's band structure, affecting the excitons' effective mass47 and their mean free time. We observed such non-uniformity after extracting the drift velocity from our measurements of the excitonic density as a function of position and time. We calculated the peak position's shift of the exciton density as a function of time, which essentially represents the drift velocity of the exciton density due to the applied strain gradient. Figure 4 shows the exciton density's peak position as a function of time when the excitation was placed just to the left of the high tensile strain point on top of the SiO2 pillar [L in Fig. 3(a)], and the mobility resulting from the local strain experienced by the density's peak. Note that the exciton density's peak position follows the direction of the strain gradient and moves towards the position of highest tensile strain as expected. With the approximate local strain profile shown in Fig. 4(a), the exciton density appears to reach drift velocities as large as 250 m/s and drift distances as large as 500 nm.

FIG. 4.

(a) Local strain profile and strain gradient experienced by the exciton density's peak of the WSe2 monolayer when the excitation was positioned just to the left of the pillar [L position in Fig. 3(a)], and the APD was scanned along the horizontal direction from left to right. (b) Exciton density's peak position and drift velocity as a function of time. Both the experimental strain and density's peak position were fit with second-order polynomials to calculate smooth strain gradient and drift velocity curves, respectively. (c) Mobility obtained from experimental drift velocity and local strain gradient. Two different units for mobility are shown to compare the strain mobility to the traditional mobility. (d) and (e) Experimental and numerical normalized exciton densities as a function of position and time. The peak of the numerical exciton density is overlain on top of the experimental density for comparison. The parameters used in the numerical solution are α=0.66,Γ=0.17,τ=0.73ns,μϵ=18cm2s1%1,Eg/ϵ=55meV/%.

FIG. 4.

(a) Local strain profile and strain gradient experienced by the exciton density's peak of the WSe2 monolayer when the excitation was positioned just to the left of the pillar [L position in Fig. 3(a)], and the APD was scanned along the horizontal direction from left to right. (b) Exciton density's peak position and drift velocity as a function of time. Both the experimental strain and density's peak position were fit with second-order polynomials to calculate smooth strain gradient and drift velocity curves, respectively. (c) Mobility obtained from experimental drift velocity and local strain gradient. Two different units for mobility are shown to compare the strain mobility to the traditional mobility. (d) and (e) Experimental and numerical normalized exciton densities as a function of position and time. The peak of the numerical exciton density is overlain on top of the experimental density for comparison. The parameters used in the numerical solution are α=0.66,Γ=0.17,τ=0.73ns,μϵ=18cm2s1%1,Eg/ϵ=55meV/%.

Close modal

Both the local strain as a function of position and the exciton density's peak position as a function of time were fit with second-order polynomials to obtain smooth strain gradient, drift velocity, and mobility curves. As expected, the resulting strain mobility appears to be non-uniform over the range where the density's peak moves. However, assuming that such variation is negligible, the model given by Eq. (1), where the mobility is assumed to be uniform, may still be used to qualitatively explain our observations. Our simulation result of the exciton density under an experimentally obtained strain profile along one spatial dimension and time is shown in Fig. 4(e). The inputs to the numerical solution were the experimentally obtained exciton density at t=0 as the initial condition, and the experimentally obtained parameters Dt, ϵx and μϵ, where the strain mobility was averaged over the spatial range where the exciton density's peak moved. As expected, the numerical exciton density's peak shifts towards the high tensile strain point on top of the pillar. While the experimental exciton density appears to become slightly asymmetric probably as the result of variations in the local strain field that our diffraction-limited imaging technique is unable to identify, the numerical and experimental peak positions are in reasonable agreement as illustrated in Fig. 4(d), indicating that the drift-diffusion-relaxation model described by Eq. (1) is able to replicate the main feature of our observations.

In summary, we have shown that the motion of excitonic energy in monolayer WSe2 can be controlled via strain gradients. The sensitivity of WSe2's direct bandgap to strain enabled the creation of local excitonic potentials that resulted in the directional drift of exciton densities towards high-tensile strain points. Furthermore, the large susceptibility of WSe2 monolayers to disorder led to anomalous diffusive transport with a time-varying diffusion coefficient. These observations are captured by the proposed drift-diffusion-relaxation model that includes the strain field on the WSe2 monolayer. The capability to control the motion of excitonic energy on monolayer TMDs with engineered strain profiles opens up the possibility to design novel, room-temperature-stable excitonic devices4,5 that prescind from the resistive-capacitive delays, heat generation, and scalability limitations that hinder conventional optoelectronic technologies.

See supplementary material for details of the substrate fabrication and its topography, additional transport and spectral data, and details of the model derivation and numerical solution.

This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1256260. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. This work was also funded through AFOSR Grant No. 16RT1256 and startup grant from the University of Michigan. D.C L. would like to acknowledge support from NSF's Graduate Research Fellowship Program and University of Michigan's Rackham Merit Fellowship. The authors would like to acknowledge the Lurie Nanofabrication Facility at the University of Michigan where the substrates were fabricated, and Kanak Datta for helpful discussions.

1.
Y. J.
Zhang
,
T.
Oka
,
R.
Suzuki
,
J. T.
Ye
, and
Y.
Iwasa
,
Science
344
,
725
(
2014
).
2.
S.-H.
Gong
,
F.
Alpeggiani
,
B.
Sciacca
,
E. C.
Garnett
, and
L.
Kuipers
,
Science
359
,
443
(
2018
).
3.
H.
Chen
,
M.
Liu
,
L.
Xu
, and
D. N.
Neshev
,
Beilstein J. Nanotechnol.
9
,
780
(
2018
).
4.
L. V. V.
Butov
,
Superlattices Microstruct.
108
,
2
(
2017
).
5.
D.
Unuchek
,
A.
Ciarrocchi
,
A.
Avsar
,
K.
Watanabe
,
T.
Taniguchi
, and
A.
Kis
,
Nature
560
,
340
(
2018
).
6.
M.-L.
Tsai
,
S.-H.
Su
,
J.-K.
Chang
,
D.-S.
Tsai
,
C.-H.
Chen
,
C.-I.
Wu
,
L.-J.
Li
,
L.-J.
Chen
, and
J.-H.
He
,
ACS Nano
8
,
8317
(
2014
).
7.
Z.
Yin
,
H.
Li
,
H.
Li
,
L.
Jiang
,
Y.
Shi
,
Y.
Sun
,
G.
Lu
,
Q.
Zhang
,
X.
Chen
, and
H.
Zhang
,
ACS Nano
6
,
74
(
2012
).
8.
S. B.
Desai
,
S. R.
Madhvapathy
,
A. B.
Sachid
,
J. P.
Llinas
,
Q.
Wang
,
G. H.
Ahn
,
G.
Pitner
,
M. J.
Kim
,
J.
Bokor
,
C.
Hu
,
H.-S. P.
Wong
, and
A.
Javey
,
Science
354
,
99
(
2016
).
9.
V.
Podzorov
,
M. E.
Gershenson
,
C.
Kloc
,
R.
Zeis
, and
E.
Bucher
,
Appl. Phys. Lett.
84
,
3301
(
2004
).
10.
O.
Lopez-Sanchez
,
D.
Lembke
,
M.
Kayci
,
A.
Radenovic
, and
A.
Kis
,
Nat. Nanotechnol.
8
,
497
(
2013
).
11.
J. S.
Ross
,
P.
Klement
,
A. M.
Jones
,
N. J.
Ghimire
,
J.
Yan
,
D. G.
Mandrus
,
T.
Taniguchi
,
K.
Watanabe
,
K.
Kitamura
,
W.
Yao
,
D. H.
Cobden
, and
X.
Xu
,
Nat. Nanotechnol.
9
,
268
(
2014
).
12.
A.
Pospischil
,
M. M.
Furchi
, and
T.
Mueller
,
Nat. Nanotechnol.
9
,
257
(
2014
).
13.
B. W. H.
Baugher
,
H. O. H.
Churchill
,
Y.
Yang
, and
P.
Jarillo-Herrero
,
Nat. Nanotechnol.
9
,
262
(
2014
).
14.
F.
Withers
,
O.
Del Pozo-Zamudio
,
A.
Mishchenko
,
A. P.
Rooney
,
A.
Gholinia
,
K.
Watanabe
,
T.
Taniguchi
,
S. J.
Haigh
,
A. K.
Geim
,
A. I.
Tartakovskii
, and
K. S.
Novoselov
,
Nat. Mater.
14
,
301
(
2015
).
15.
C.
Zhang
,
A.
Johnson
,
C. L.
Hsu
,
L. J.
Li
, and
C. K.
Shih
,
Nano Lett.
14
,
2443
(
2014
).
16.
M. M.
Ugeda
,
A. J.
Bradley
,
S.-F.
Shi
,
F. H.
da Jornada
,
Y.
Zhang
,
D. Y.
Qiu
,
W.
Ruan
,
S.-K.
Mo
,
Z.
Hussain
,
Z.-X.
Shen
,
F.
Wang
,
S. G.
Louie
, and
M. F.
Crommie
,
Nat. Mater.
13
,
1091
(
2014
).
17.
K.
He
,
N.
Kumar
,
L.
Zhao
,
Z.
Wang
,
K. F.
Mak
,
H.
Zhao
, and
J.
Shan
,
Phys. Rev. Lett.
113
,
026803
(
2014
).
18.
T. C.
Berkelbach
,
M. S.
Hybertsen
, and
D. R.
Reichman
,
Phys. Rev. B
88
,
045318
(
2013
).
19.
M.
Chhowalla
,
H. S.
Shin
,
G.
Eda
,
L. J.
Li
,
K. P.
Loh
, and
H.
Zhang
,
Nat. Chem.
5
,
263
(
2013
).
20.
C. H.
Chang
,
X.
Fan
,
S. H.
Lin
, and
J. L.
Kuo
,
Phys. Rev. B-Condens. Matter Mater. Phys.
88
,
195420
(
2013
).
21.
P.
Lu
,
X.
Wu
,
W.
Guo
, and
X. C.
Zeng
,
Phys. Chem. Chem. Phys.
14
,
13035
(
2012
).
22.
P.
Johari
and
V. B.
Shenoy
,
ACS Nano
6
,
5449
(
2012
).
23.
H. J.
Conley
,
B.
Wang
,
J. I.
Ziegler
,
R. F.
Haglund
,
S. T.
Pantelides
, and
K. I.
Bolotin
,
Nano Lett.
13
,
3626
(
2013
).
24.
S. B.
Desai
,
G.
Seol
,
J. S.
Kang
,
H.
Fang
,
C.
Battaglia
,
R.
Kapadia
,
J. W.
Ager
,
J.
Guo
, and
A.
Javey
,
Nano Lett.
14
,
4592
(
2014
).
25.
J.
Feng
,
X.
Qian
,
C.-W. W.
Huang
, and
J.
Li
,
Nat. Photonics
6
,
866
(
2012
).
26.
N.
Kumar
,
Q.
Cui
,
F.
Ceballos
,
D.
He
,
Y.
Wang
, and
H.
Zhao
,
Nanoscale
6
,
4915
(
2014
).
27.
M.
Kulig
,
J.
Zipfel
,
P.
Nagler
,
S.
Blanter
,
C.
Schüller
,
T.
Korn
,
N.
Paradiso
,
M. M.
Glazov
, and
A.
Chernikov
,
Phys. Rev. Lett.
120
,
207401
(
2018
).
28.
F.
Cadiz
,
C.
Robert
,
E.
Courtade
,
M.
Manca
,
L.
Martinelli
,
T.
Taniguchi
,
K.
Watanabe
,
T.
Amand
,
A. C. H.
Rowe
,
D.
Paget
,
B.
Urbaszek
, and
X.
Marie
,
Appl. Phys. Lett.
112
,
152106
(
2018
).
29.
M.
Newburger
,
E.
McCormick
,
J.
Xu
, and
R.
Kawakami
, presented at the APS March Meeting 2018, Los Angeles, California, 5–9 March 2018, abstract L21.00008, available at https://meetings.aps.org/Meeting/MAR18/Session/L21.8.
30.
T.
Kato
and
T.
Kaneko
,
ACS Nano
10
,
9687
(
2016
).
31.
K. S.
Novoselov
,
D.
Jiang
,
F.
Schedin
,
T. J.
Booth
,
V. V.
Khotkevich
,
S. V.
Morozov
, and
A. K.
Geim
,
Proc. Natl. Acad. Sci. U. S. A.
102
,
10451
(
2005
).
32.
W. H.
Chae
,
J. D.
Cain
,
E. D.
Hanson
,
A. A.
Murthy
, and
V. P.
Dravid
,
Appl. Phys. Lett.
111
,
143106
(
2017
).
33.
P. B.
Deotare
,
W.
Chang
,
E.
Hontz
,
D. N.
Congreve
,
L.
Shi
,
P. D.
Reusswig
,
B.
Modtland
,
M. E.
Bahlke
,
C. K.
Lee
,
A. P.
Willard
,
V.
Bulović
,
T.
Van Voorhis
, and
M. A.
Baldo
,
Nat. Mater.
14
,
1130
(
2015
).
34.
G. M.
Akselrod
,
P. B.
Deotare
,
N. J.
Thompson
,
J.
Lee
,
W. A.
Tisdale
,
M. A.
Baldo
,
V. M.
Menon
, and
V.
Bulović
,
Nat. Commun.
5
,
3646
(
2014
).
35.
J.
Wu
and
K. M.
Berland
,
Biophys. J.
95
,
2049
(
2008
).
36.
J.-P.
Bouchaud
and
A.
Georges
,
Phys. Rep.
195
,
127
(
1990
).
37.
G.
Zumofen
,
J.
Klafter
, and
A.
Blumen
,
J. Stat. Phys.
65
,
991
(
1991
).
38.
S.
Havlin
and
D.
Ben-Avraham
,
Adv. Phys.
36
,
695
(
1987
).
39.
R.
Metzler
and
J.
Klafter
,
J. Phys. A: Math. Gen.
37
,
R161
(
2004
).
40.
Z.
Lin
,
B. R.
Carvalho
,
E.
Kahn
,
R.
Lv
,
R.
Rao
,
H.
Terrones
,
M. A.
Pimenta
, and
M.
Terrones
,
2D Mater.
3
,
022002
(
2016
).
41.
L.
Yuan
,
T.
Wang
,
T.
Zhu
,
M.
Zhou
, and
L.
Huang
,
J. Phys. Chem. Lett.
8
,
3371
(
2017
).
42.
M.
Buscema
,
G. A.
Steele
,
H. S. J.
van der Zant
, and
A.
Castellanos-Gomez
,
Nano Res.
7
,
561
(
2014
).
43.
X.
Fu
,
G.
Jacopin
,
M.
Shahmohammadi
,
R.
Liu
,
M.
Benameur
,
J. D.
Ganière
,
J.
Feng
,
W.
Guo
,
Z. M.
Liao
,
B.
Deveaud
, and
D.
Yu
,
ACS Nano
8
,
3412
(
2014
).
44.
K. F.
Brennan
,
The Physics of Semiconductors with Applications to Optoelectronic Devices
(
Cambridge University Press
,
Cambridge
,
1999
).
45.
J.
Singh
,
Electronic and Optoelectronic Properties of Semiconductor Structures
(
Cambridge University Press
,
2003
).
46.
J. P.
Wolfe
,
Phys. Today
35
(
3
),
2463
(
1982
).
47.
L.
Dong
,
R. R.
Namburu
,
T. P.
O'Regan
,
M.
Dubey
, and
A. M.
Dongare
,
J. Mater. Sci.
49
,
6762
6771
(
2014
).

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