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Showcasing research from the Beijing Computational Science Research Center, Northwestern University, USA, the University of Science and Technology of China, the Bremen Center for Computational Materials Science, and the University of Hong Kong .

Quantum mechanical modeling the emission pattern and

polarization of nanoscale light emitting diodes

A quantum mechanical method for modeling electroluminescence processes in nanoscale light emitting diodes has been developed. Interactions with the electromagnetic vacuum environment are included within the non-equilibrium Green’s function framework. The emission spectrum of a realistic silicon nanowire light emitting diode was simulated and its emission characteristics were determined. This work makes an important step forward towards atomistic modeling of electrically induced optical response in nanoscale systems.

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Nanoscalewww.rsc.org/nanoscale

ISSN 2040-3364

REVIEW ARTICLE Ewa M. Goldys et al. Lanthanide upconversion luminescence at the nanoscale: fundamentals and optical properties

Volume 8 Number 27 21 July 2016 Pages 13083–13524

As featured in:

See ChiYung Yam et al. , Nanoscale , 2016, 8 , 13168.

Nanoscale

COMMUNICATION

Cite this: Nanoscale, 2016, 8, 13168

Received 21st March 2016,Accepted 20th May 2016

DOI: 10.1039/c6nr02356h

www.rsc.org/nanoscale

Quantum mechanical modeling the emissionpattern and polarization of nanoscale lightemitting diodes

Rulin Wang,a Yu Zhang,b Fuzhen Bi,a,c Thomas Frauenheim,d GuanHua Chene andChiYung Yam*a,e

Understanding of the electroluminescence (EL) mechanism in

optoelectronic devices is imperative for further optimization of

their efficiency and effectiveness. Here, a quantum mechanical

approach is formulated for modeling the EL processes in nanoscale

light emitting diodes (LED). Based on non-equilibrium Green’s

function quantum transport equations, interactions with the

electromagnetic vacuum environment are included to describe

electrically driven light emission in the devices. The presented

framework is illustrated by numerical simulations of a silicon

nanowire LED device. EL spectra of the nanowire device under

different bias voltages are obtained and, more importantly, the

radiation pattern and polarization of optical emission can be

determined using the current approach. This work is an important

step forward towards atomistic quantum mechanical modeling

of the electrically induced optical response in nanoscale

systems.

Electroluminescence (EL) is an important phenomenonemployed in light emitting diode (LED) technology wherelight is emitted from a solid state material in response to anelectrical power source. Much work has been devoted to thedevelopment of LED technology that has led to continuousadvancements in both efficiencies and optical power.1 Newefforts are now directed to exploit semiconductor nano-structures that exhibit extraordinary optical and electronic pro-perties. A more ambitious use of nanostructure devices isto exploit quantum effects which fundamentally change themechanism of electrical-to-optical power conversion. Thesedevices are made possible with the continuous development ofnanofabrication techniques and are emerging as promising

candidates for optoelectronic and energy devices. Indeed,electrically driven light emission has been reported from asingle carbon nanotube and nanowire,2–5 monolayer transitionmetal dichalcogenides6,7 and, to the ultimate miniaturizationlimit, from a single molecule.8,9

Understanding the EL mechanism in nanoscale LEDdevices is crucial to further advance the technology for moreefficient lighting and enhanced communications. From thetheoretical perspective, accurate description of the electrical-to-optical conversion processes is a challenging task, since thesystem is in the non-equilibrium state driven by an optical andelectric field. In this context, atomic level modeling is becom-ing increasingly relevant, not only for accurate description ofthe coupled optical–electrical processes, but also to cope withthe myriad architectures and chemical compositions inmodern devices. The prevailing studies evaluate the perform-ance of LED devices based on classical models, relyingon parameters obtained either from experiments10,11 orfirst-principles calculations.12 However, these models fail tocapture quantum phenomena and break down at the nano-scale. For microscopic systems, optical emission has beenstudied using Fermi’s golden rule (FGR) to evaluate transitionrates between energy levels.13–15 The first attempt to includequantum effects to directly simulate the EL process was madeby Galperin et al. for model systems.16,17 Recently, a diagram-matic approach has been formulated to study EL in molecularjunctions.18,19 In this letter, we present a quantum mechanicalmethod for realistic LED device simulations. Taking intoaccount the atomistic details and non-equilibrium statistics,EL spectra of nanoscale devices under different bias con-ditions can be simulated. Furthermore, the method offersthe possibility of determining the radiation pattern andpolarization of emitted light.20 This provides useful in-formation for further improvement of the device perform-ance21 as well as probing structural details in thejunctions.22,23

Quantum transport approaches based on the non-equilibrium Green’s function (NEGF) method provide an

aBeijing Computational Science Research Center, Haidian District, Beijing 100193,

China. E-mail: yamcy@csrc.ac.cnbDepartment of Chemistry, Northwestern University, Evanston, Illinois 60208, USAcUniversity of Science and Technology of China, Hefei, Anhui 230026, ChinadBremen Center for Computational Materials Science, Am Fallturm 1a,

D-28359 Bremen, GermanyeDepartment of Chemistry, The University of Hong Kong, Pokfulam Road, Hong Kong,

China

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efficient and versatile way to describe the coupled optical–electrical processes in nanoscale devices.24–27 Based on theKeldysh NEGF approach, steady state current can be obtainedfrom28

Iα ¼ 2eℏ

ðdE2π

Tr Σ,α Eð ÞG> Eð Þ � Σ>

α Eð ÞG, Eð Þ� � ð1Þ

where G<,> are lesser and greater Green’s functions, providinginformation on the energy states and population statistics forelectrons and holes, respectively. Σ,;.

α are the self-energiesand α corresponds to a particular scattering process. Consider-ing a two-terminal LED device, the scattering processes arisefrom the contacts and also electron–photon interaction. Thefirst and second terms in the square bracket of eqn (1) areinterpreted respectively as the incoming and outgoing rate ofelectrons in the device due to the scattering processes. Thus,Iα gives the steady state current resulting from different scatter-ing processes.

The self-energy associated with the contacts can beobtained following the standard procedure,29 whereas theexplicit evaluation of electron–photon self-energy, Σ,;.

ep

requires many body diagrammatic techniques and its self-consistent Born approximation (SCBA) expression is givenby30,31

Σ,;.ep Eð Þ ¼

Xq

Mq Nq þ 1� �

G,;. E+ ℏωq� ��

þ NqG,;. E+ ℏωq� ��

Mq

ð2Þ

where Nq is the photon occupation number and ωq is thephoton frequency. q refers to photon mode characterized byits wave vector ~kq and polarization directions ~λq. The threevectors are mutually perpendicular to one another and aredefined as

~k ¼ sin θ cos ϕ; sin θ sin ϕ; cos θð Þ~λq;k ¼ sin ϕ;� cos ϕ; 0ð Þ~λq;? ¼ cos θ cos ϕ; cos θ sin ϕ; � sin θð Þ

8><>: ð3Þ

For EL processes, the associated self-energy accounts forinteractions with electromagnetic field modes in their vacuumstate (Nq = 0). The system then undergoes spontaneous emis-sion by relaxation to a lower energy state. Σ,;.

ep for spontaneousemission is thus given by

Σ,;.ep Eð Þ ¼

Xq

MqG,;. E+ ℏωq� �

Mq ð4Þ

Here, Mq is the electron–photon coupling matrix and itselements are given by24,25

Mq;μν ¼ em

ℏ2ε0ωqV

� �1=2~λq � μh j~p νj i ð5Þ

where ℏ is the reduced Planck constant; ε0 is the vacuum per-mittivity; V is the volume; μ and ν correspond to atomic basis

functions. The infinite sum in eqn (4) is transformed tointegration

Σ,;.ep Eð Þ ¼

ð10d ℏωð ÞΣ,;.

ep E;ωð Þ

¼ð10d ℏωð Þ

ðπ0dθ sin θ

ð2π0dϕ

� Σ,;.k E;ω; θ;ϕð Þ þ Σ,;.

? E;ω; θ;ϕð Þh i

ð6Þ

where Σ,;.k E;ω; θ;ϕð Þ and Σ,;.

? E;ω; θ;ϕð Þ are defined as angle-dispersed self-energies for the two perpendicular polarizationdirections,

Σ,;.k E;ω; θ;ϕð Þ ¼R,;.

xx sin2 ϕþ R,;.yy cos2 ϕ

� R,;.xy þ R,;.

yx

� sin ϕ cos ϕ

ð7Þ

Σ,;.? E;ω; θ;ϕð Þ ¼ ½R,;.

xx cos2 ϕþ R,;.xy þ R,;.

yx

� cos ϕ sin ϕ

þ R,;.yy sin2 ϕ� cos2 θ þ R,;.

zz sin2 θ

� R,;.xz þ R,;.

zx

� �cos ϕ cos θ sin θ

� R,;.yz þ R,;.

zy

� sin ϕ cos θ sin θ

ð8Þand

R,;.ij ¼ PiG,;. E+ ℏωð ÞPj

Pi;μν ¼ ωe2

16π3c3m2ε0

� �1=2

μh jpi νj ið9Þ

and i, j ∈ (x, y, z). The Green’s function in eqn (1) can then beobtained from the Keldysh equation

G,;. Eð Þ ¼Xα

Gr Eð ÞΣ,;.α Eð ÞGa Eð Þ: ð10Þ

where Gr and Ga are retarded and advanced Green’s functions.Substituting eqn (6) into eqn (1), we obtain the steady state

current due to electron–photon scattering, Iep, which shouldbe zero since the number of electrons should be conservedduring emission of photons. The first (second) term in eqn (1)corresponds to the transition of an electron from the energylevel E + ℏω (E) to E (E − ℏω) while emitting a photon withenergy ℏω. Thus, the emission flux Fem for photon frequencyω can be expressed as

Fem ωð Þ ¼ 2ℏ

ðdE2π

Tr Σ,ep E;ωð ÞG> Eð Þ

h ið11Þ

More importantly, the radiation pattern and polarization ofemitted photons can be determined by substituting the angle-dispersed self-energies eqn (7) and (8) into eqn (11).

Femk ω; θ;ϕð Þ ¼ 2

ℏ

ðdE2π

Tr Σ,k E;ω; θ;ϕð ÞG> Eð Þ

h i

Fem? ω; θ;ϕð Þ ¼ 2

ℏ

ðdE2π

Tr Σ,? E;ω; θ;ϕð ÞG> Eð Þ� � ð12Þ

FGR has been commonly used to evaluate the rate of spon-taneous emissions. For simple two-level systems, eqn (11)

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recovers the FGR rate expression for electron transitionbetween the levels. It is however important to emphasize thatin the current approach, the electronic structure of the deviceunder non-equilibrium conditions is explicitly calculated. Inparticular, the occupations of electronic states due to non-equilibrium statistics are taken into account.

We apply the method to model a nanoscale LED devicebased on a silicon nanowire with a cross section diameter of1.5 nm. The nanowire is 9.5 nm in length oriented in the [110]direction. Silicon is recognized as an inefficient light-emitterdue to its indirect band gap. However, due to quantum con-finement effects, a transition from an indirect to a direct bandgap occurs for silicon nanowires with sufficiently small dimen-sions. The previous theoretical study based on the DFTBmethod reported that the band gaps of silicon nanowiresremain direct even for diameters up to 7 nm.32 Recentadvancements in nanotechnology have shown that the deviceperformance based on silicon nanostructures can approach oreven exceed that of equivalent direct band gap materials.33

Experimentally, EL in silicon nanowires with a diameter downto 5 nm has been observed.34 An atomistic model is employedin the current study which contains 1000 atoms. To form ap–n junction, Ga and As atoms are explicitly doped in the systemto give a doping concentration of about 2.0 × 1020 cm−3. Thesurface of the nanowire is passivated with hydrogen atoms toeliminate dangling bonds. The device is connected to twosemi-infinite doped silicon leads where an external biasvoltage is applied. The electronic structure of the model isdescribed at the density functional tight-binding (DFTB)level35,36 and a pbc-0–3 parameter set is used.37 At equili-brium, an internal built-in voltage Vbi of 2.44 V, which is thedifference between the Fermi levels of the left and right leads,is formed across the two different doped regions. The simu-lations are performed at 300 K.

We solve eqn (11) to obtain EL spectra of the nanowiredevice under different external bias voltages. In this work, thelowest order expansion to the self-energy Σ,;.

ep is employed.Physically, this corresponds to the situation where electron dis-tribution in the device is unaffected by electron–photon inter-action. This can be justified by the fact that the interactionwith the electromagnetic vacuum environment is weak. There-fore, the electronic structure remains intact and nonlineareffects are neglected. Fig. 1 plots the local density of states(LDOS) of the device along the wire direction for forward biasvoltages of (a) 1.0 V and (b) 2.6 V, together with the electrondistribution ρe(E) in the conduction band and hole distri-bution ρh(E) in the valence band,

ρe Eð Þ ¼ 12π

Im Tr G, Eð ÞS� � �;

ρh Eð Þ ¼ � 12π

Im Tr G> Eð ÞS½ �f gð13Þ

where S is the overlap matrix. Clearly, a built-in voltage isformed across the junction, as shown in Fig. 1(a). Due to thispotential barrier, electrons are localized at the n-doped regionwhile holes are localized at the p-doped region. The electron–

hole recombination is inhibited and the emission processis suppressed in this case. When the forward bias is appliedto the device, the potential difference across the junctionis reduced. As shown in Fig. 1(b), conducting channelsare formed at the conduction band and valence band edgesfor electrons and holes, respectively. The carriers can thenmove along the channels driven by the external bias voltage.Due to their spatial proximity, the electron–hole pairsundergo a recombination and energy is emitted in the form ofphotons.

EL spectra of the nanowire LED device is plotted in Fig. 2for different bias voltages. A single broad emission peak isobserved corresponding to transitions from the conductionband to the valence band. This is in contrast to that of mole-cular junctions19 where multiple peaks are observed due tomolecular resonances. The shape of the emission peak isasymmetric with a tail at the higher energy side due to theFermi–Dirac distribution of charge carriers. We note that theintensity of photon emission in general increases with anapplied bias voltage. For a bias voltage below 2.0 V, no lightemission is observed. This is consistent with the results shownin the LDOS, where electron–hole recombination is suppressedwhen the applied bias voltage is lower than the internal built-in voltage of the device. As the forward bias approaches theflat band position, electrons and holes are injected simul-taneously from electrodes. For carriers injected into thedevice, they can either flow directly from source to drain elec-trodes without losing energy or undergo inelastic scatteringthrough electron–photon interactions. The latter results inradiative recombination and gives rise to photon emission.

Fig. 1 LDOS of the device along the nanowire axis for a forward biasvoltage of (a) 1.0 V and (b) 2.6 V. The left side of the nanowire isp-doped and the right side is n-doped. A built-in potential is formed atthe junction due to the space charge. Grey line in (b): local emissionalong the wire direction for a photon frequency of 2.55 eV. Right: elec-tron (red) and hole (blue) distributions versus energy for the two appliedbias voltages.

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The emission intensity therefore increases substantially whenthe applied bias exceeds the built-in potential of the system, asshown in Fig. 2. For a bias voltage of 2.6 V, a strong EL peak isobserved at a light frequency of 2.55 eV. The frequency ofemitted photons is mainly determined by DOS and the elec-tron distribution in the conduction and valence bands. Ingeneral, charge carriers relax non-radiatively as they passthrough the device and result in near band edge emission.For nanoscale devices, energy level spacing between electronstates increases due to quantum confinement. In addition,the system studied in this work is small compared to thecoherence length.38 Therefore, the nonradiative decay due tophonon scattering is expected to be suppressed. Phonon scat-tering can be included similarly as eqn (2) within NEGFformalism39–41 and its effect on EL of the nanoscale deviceneeds further investigations. From eqn (11), we can determinelocal emission by decomposing the contributions to emissionfrom individual atoms. The local emission flux along thenanowire direction is plotted in Fig. 1(b). We observe thatphotons are emitted mainly from the p-doped region. Due todifferent doping atoms, the electronic structures of p-dopedand n-doped regions are different. In Fig. 1, it is observedthat DOS near the valence band edge of the n-doped regionand the conduction band edge of the p-doped region aredecreased. This lowering of DOS results in a reduction ofminority carriers in the corresponding regions. Since the DOSnear the valence band edge of the n-doped region is decreasedto a larger extent, the corresponding electron–hole recombina-tion and hence the emission is reduced. This is in agreementwith previous experiments that charge carriers in p-dopedsilicon have a higher recombination rate compared to n-dopedsilicon.42

The optical emission from the nanowire LED device isfurther characterized by its propagation and polarization direc-tions. Eqn (12) allows analysis of its spatial distribution alongthe two polarization vectors. Fig. 3(a) shows the EL radiation

pattern of the silicon LED device under a bias voltage of 2.4 V.The emitted light frequency is chosen as 2.4 eV. The siliconnanowire is oriented along the x-axis. The key feature we notein Fig. 3(a) is that light is emitted from the surface of thenanowire and essentially no edge emission is observed. Wefurther analyse the polarization of emitted light in Fig. 3(b).The blue line gives the polar plot of the emission flux in they–z plane while the green line plots in the x–z plane. Femk andFem⊥ correspond to emitted photons with in-plane and out-of-plane polarizations, respectively. Here, θ is defined as theangle measured from the z-axis. In the y–z plane (blue line),the polarization direction of Femk is perpendicular to the nano-wire axis and the corresponding emission flux is zero for all θ.In contrast, Fem⊥ is aligned along the x-direction which is paral-lel to the nanowire device and we observe a constant emissionflux with respect to θ. The result clearly shows that the silicon

Fig. 3 (a) EL radiation pattern of the silicon LED device under a 2.4 Vforward bias voltage. Light frequency is set as 2.4 eV. The color rep-resents the intensity of emitted light and the angular coordinates corres-pond to the propagation direction. (b) Polar plot of EL intensity as afunction of θ. Blue line: Femk and Fem⊥ in the y–z plane; green line: Femk andFem⊥ in the x–z plane.

Fig. 2 Electroluminescence spectrum of the silicon nanowire LEDdevice for various forward bias voltages. Grey line: 2.2 V; red line: 2.3 V;blue line: 2.4 V; green line: 2.5 V; black line: 2.6 V.

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nanowire LED behaves as a linearly polarized radiation sourcewhere the emitted light is polarized along the nanowire deviceaxis. This is consistent with the experimental observation oflight emission from quasi-one-dimensional devices.43,44 In thex–z plane (green line), Fem⊥ is instead y-polarized and noemitted light polarized in the y-direction is observed. On theother hand, the polarization direction of Femk makes an angle θ

with respect to the nanowire axis. As shown in Fig. 3(b), Femk isproportional to cos2 θ, giving maximum EL intensity when it isaligned parallel to the nanowire axis. This can be explainedby the fact that in the weak coupling limit, electron–photoncoupling is proportional to ~A �~p, where ~A is aligned along thepolarization direction and ~p is in general aligned alongthe direction of current flow. Consequently, the emission fluxFemk in the x–z plane is proportional to cos2 θ.

In conclusion, we formulate a quantum mechanicalapproach for modeling nanoscale LED devices based on NEGFquantum transport formalism. The non-equilibrium statisticsin the device due to applied voltage and interactions with lightare taken into account and EL processes in LED devices can beaccurately described. The current approach provides the toolsfor determining not only the spectrum but also the radiationpattern and polarization of optical emission in nanoscaledevices. We demonstrate the method by simulations of ELproperties of a silicon nanowire LED device.

Optoelectronics has become an integral part of our lives.Given the complexity of modern nanoscale devices, atomisticdetails and quantum effects are playing increasingly key rolesin determining the device properties. Important also is tounderstand the EL properties of single molecules in scanningtunneling microscopy experiments.45,46 By theoretical predic-tions of an emission pattern, information on the molecularstructure and dynamics can be elucidated. This sheds light onthe structural information of single molecules in junctionswhich is one of the fundamental issues in the field of mole-cular electronics. The quantum mechanical method presentedin this work provides an efficient research tool for theoreticalstudies of coupled optical–electrical processes in these nano-scale systems. It should be useful for understanding the pro-perties of existing devices as well as the engineering andrational design of novel devices that meet ever-increasingperformance demands.

Acknowledgements

The financial support from the National Natural Science Foun-dation of China (21322306(C. Y. Y.), 21273186(G. H. C.,C. Y. Y.)), the National Basic Research Program of China (No.2014CB921402 (C. Y. Y.)), and the University Grant Council(AoE/P-04/08(G. H. C., C. Y. Y.)) is gratefully acknowledged. Theauthors would like to thank Shiwu Gao and Wen Yang forhelpful discussions. The computational support from theSpecial Program for Applied Research on Super Computationof the NSFC-Guangdong Joint Fund (the second phase) and

the Beijing Computational Science Research Center (CSRC) isalso acknowledged.

References

1 E. F. Schubert, T. Gessmann and J. K. Kim, Light emittingdiodes, Wiley Online Library, 2005.

2 T. Mueller, M. Kinoshita, M. Steiner, V. Perebeinos,A. A. Bol, D. B. Farmer and P. Avouris, Nat. Nanotechnol.,2010, 5, 27.

3 Y. Huang and C. M. Lieber, Pure Appl. Chem., 2004, 76,2051.

4 J. Bao, M. A. Zimmler, F. Capasso, X. Wang and Z. F. Ren,Nano Lett., 2006, 6, 1719.

5 E. D. Minot, F. Kelkensberg, M. V. Kouwen, J. A. V. Dam,L. P. Kouwenhoven, V. Zwiller, M. T. Borgström,O. Wunnicke, M. A. Verheijen and E. P. A. M. Bakkers,Nano Lett., 2007, 7, 367.

6 R. S. Sundaram, M. Engel, A. Lombardo, R. Krupke,A. C. Ferrari, P. Avouris and M. Steiner, Nano Lett., 2013,13, 1416.

7 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., 2014,9, 268.

8 C. W. Marquardt, S. Grunder, A. Baszczyk, S. Dehm,F. Hennrich, H. V. Löhneysen, M. Mayor and R. Krupke,Nat. Nanotechnol., 2010, 5, 863.

9 G. Reecht, F. Scheurer, V. Speisser, Y. Dappe, F. Mathevetand G. Schull, Phys. Rev. Lett., 2014, 112, 047403.

10 M.-H. Kim, M. F. Schubert, Q. Dai, J. K. Kim,E. F. Schubert, J. Piprek and Y. Park, Appl. Phys. Lett., 2007,91, 183507.

11 G. G. Malliaras and J. C. Scott, J. Appl. Phys., 1999, 85, 7426.12 P. Kordt, J. J. M. van der Holst, M. A. Helwi, W. Kowalsky,

F. May, A. Badinski, C. Lennartz and D. Andrienko, Adv.Funct. Mater., 2015, 25, 1955.

13 A. F. V. Driel, G. Allan, C. Delerue, P. Lodahl, W. L. Vos andD. Vanmaekelbergh, Phys. Rev. Lett., 2005, 95, 236804.

14 G. Tian, J.-C. Liu and Y. Luo, Phys. Rev. Lett., 2011, 106,177401.

15 D. Shiri, A. Verma, C. R. Selvakumar and M. P. Anantram,Sci. Rep., 2012, 2, 461.

16 M. Galperin and A. Nitzan, Phys. Rev. Lett., 2005, 95,206802.

17 M. Galperin and A. Nitzan, Phys. Chem. Chem. Phys., 2012,14, 9421.

18 U. Harbola, B. K. Agarwalla and S. Mukamel, J. Chem.Phys., 2014, 141, 074107.

19 H. P. Goswami, W. Hua, Y. Zhang, S. Mukamel andU. Harbola, J. Chem. Theory Comput., 2015, 11, 4304.

20 J. Motohisa, Y. Kohashi and S. Maeda, Nano Lett., 2014, 14,3653.

21 J. J. Wierer, A. David and M. M. Megens, Nat. Photonics,2009, 3, 163.

Communication Nanoscale

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ishe

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vers

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f H

ong

Kon

g L

ibra

ries

on

08/0

3/20

18 0

5:44

:02.

View Article Online

22 M. Lieb, J. Zavislan and L. Novotny, J. Opt. Soc. Am. B, 2004,21, 1210–1215.

23 E. Le Moal, S. Marguet, B. Rogez, S. Mukherjee, P. DosSantos, E. Boer-Duchemin, G. Comtet and G. Dujardin,Nano Lett., 2013, 13, 4198–4205.

24 L. E. Henrickson, J. Appl. Phys., 2002, 91, 6273.25 Y. Zhang, L. Y. Meng, C. Y. Yam and G. H. Chen, J. Phys.

Chem. Lett., 2014, 5, 1272.26 L. Meng, C. Y. Yam, Y. Zhang, R. Wang and G. H. Chen,

J. Phys. Chem. Lett., 2015, 6, 4410.27 C. Y. Yam, L. Meng, Y. Zhang and G. H. Chen, Chem. Soc.

Rev., 2015, 44, 1763.28 Y. Meir and N. S. Wingreen, Phys. Rev. Lett., 1992, 68, 2512.29 Y. Xue, S. Datta and M. A. Ratner, Chem. Phys., 2002, 281,

151.30 T. Frederiksen, M. Paulsson, M. Brandbyge and A.-P. Jauho,

Phys. Rev. B: Condens. Matter, 2007, 75, 205413.31 Y. Zhang, C. Y. Yam and G. H. Chen, J. Chem. Phys., 2013,

138, 164121.32 D. Yao, G. Zhang and B. Li, Nano Lett., 2008, 8, 4557–4561.33 F. Priolo, T. Gregorkiewicz, M. Galli and T. F. Krauss, Nat.

Nanotechnol., 2014, 9, 19–32.34 A. Irrera, P. Artoni, F. Iacona, E. F. Pecora, G. Franzo,

M. Galli, B. Fazio, S. Boninelli and F. Priolo, Nanotechno-logy, 2012, 23, 075204.

35 D. Porezag, T. Frauenheim, T. Köhler, G. Seifert andR. Kaschner, Phys. Rev. B: Condens. Matter, 1995, 51, 12947.

36 M. Elstner, D. Porezag, G. Jungnickel, J. Elsner, M. Haugk,T. Frauenheim, S. Suhai and G. Seifert, Phys. Rev. B:Condens. Matter, 1998, 58, 7260.

37 A. Sieck, D. Porezag, T. Frauenheim, M. R. Pederson andK. Jackson, Phys. Rev. A, 1997, 56, 4890.

38 W. Lu, J. Xiang, B. P. Timko, Y. Wu and C. M. Lieber, Proc.Natl. Acad. Sci. U. S. A., 2005, 102, 10046.

39 M. Galperin, M. A. Ratner and A. Nitzan, J. Phys.: Condens.Matter, 2007, 19, 103201.

40 A. Pecchia, G. Romano and A. D. Carlo, Phys. Rev. B:Condens. Matter, 2007, 75, 035401.

41 Y. Dubi and M. D. Ventra, Rev. Mod. Phys., 2011, 83, 131.42 J. E. Cotter, J. H. Guo, P. J. Cousins, M. D. Abbott,

F. W. Chen and K. C. Fisher, IEEE Trans. Electron Devices,2006, 53, 1893.

43 J. Wang, M. S. Gudiksen, X. Duan, Y. Cui and C. M. Lieber,Science, 2001, 293, 1455.

44 J. A. Misewich, R. Martel, P. Avouris, J. C. Tsang, S. Heinzeand J. Tersoff, Science, 2003, 300, 783.

45 R. Berndt, R. Gaisch, J. K. Gimzewski, B. Reihl,R. R. Schlittler, W. D. Schneider and M. Tschudy, Science,1993, 262, 1425.

46 S. W. Wu, N. Ogawa and W. Ho, Science, 2006, 312, 1362.

Nanoscale Communication

This journal is © The Royal Society of Chemistry 2016 Nanoscale, 2016, 8, 13168–13173 | 13173

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