Multi-scale modeling of 2D-material based devices

(1)

Università di Pisa

Dottorato di ricerca

in Ingegneria dell’ Informazione

Multi-scale modeling of

2D-material based devices

Doctoral Thesis

Author:

Marta Perucchini

Supervisors:

Prof. Gianluca Fiori

Prof. Giuseppe Iannaccone

Reviewers:

Prof. Francesco Driussi

Prof. Philippe Dollfus

The Coordinator of the PhD Program:

Prof. Fulvio Gini

Dipartimento di Ingegneria dell’Informazione

Pisa, November 2019

Cycle XXXII

(2)

(3)

Preface

I would like to thank Prof. Gianluca Fiori for giving me the opportunity of pursuing my Ph.D. in his group. For me, he has been a real example of hardworking and ded-ication to research and always encouraged me to improve myself. His valuable and constant advice has been fundamental for the development of this work and those included in it. I would also like to thank Prof. Giuseppe Iannaccone for his precious suggestions and inspiring talks. I would like to express my gratitude also to Prof. Driussi and Prof. Dollfus who reviewed this thesis and whose remarks have strongly improved its quality.

Special thanks go to all my colleagues, especially Teresa, Damiano and Enrique who first welcomed me and taught me most of what I know. Thank you for your patience, kindness and support: I owe you a lot. I would also like to thank the Ph.D. students with whom I shared this journey and a lot of laughs. Nino, Alessandro and Gabriele: thanks to you with the thesis comes also a C1 certificate in "Pisano".

Last but not least I would like to thank my boyfriend Lorenzo, my family and friends for the love they unconditionally gave me and the understanding they had for my being far away. You know you are what really matters in my life.

(4)

(5)

v

(6)

(7)

Abstract

In a fast-pacing world such as that of semiconductors, the fear of approaching physi-cal or technologiphysi-cal limits to the continuous sphysi-caling of electronic devices is ceaseless. Two-dimensional (2D) materials were selected for potential candidates as channel materials thanks to their atomic thickness, able to enhance the electrostatic control of the gate, reducing short-channel effects. Several bottlenecks have however been hindering the successful integration of 2D materials such as the lack of mature and appropriate technology for their production as well as a full theoretical framework able to understand and predict the behavior of 2D-material based devices. It is in fact of paramount importance to guide experimental realization through a simulation approach able to identify in advance the most effective path to follow among the countless options available. In this thesis, we provide insights for the fabrication of new devices through the multi-scale simulation method. It combines different levels of abstraction and allows great flexibility so that it can be adapted for the system under study, assuring a good trade-off between accuracy and computational cost. We first examine the ultimate performance of MoS2-channel FETs with a gate length of

1 nm for uniformly scaled devices with channel lengths in the nm range, as would be required in integrated circuits. We also evaluate the effect of the finite density of states of a carbon nanotube gate on the loss of device performance, concluding that its limited capacitive coupling poses severe limitations on the operation of the device for the presence of tunneling currents and gate-drain interactions. Secondly, we investigate the possibility of reducing the metal-semiconductor contact resistance by building lateral heterostructure (LHs) based on noble transition metal dichalco-genides. Their strong band gap dependence upon the number of layers can be used to tune the behavior from metallic to semiconducting. We show that in the case of PdS2 it is possible to achieve sub-60 mV/decade subthreshold swing thanks to

energy-filtering mechanisms due to a particular source density of states. Besides, we predict the possibility of using 2D LHs to obtain a resonant tunneling diode, with a pronounced peak-to-valley ratio of the current-voltage characteristics. Finally, we describe a drift-diffusion based method to study electronic transport in inkjet-printed 2D material networks. As a first step, we define MoS2 and graphene network

struc-tures in the 3D space, tuning geometric parameters such as flake density, size, distri-bution and shape. We obtain the in-plane and out-of-plane mobility values from ab initio simulations and solving a self-consistent algorithm we obtain the output and transfer characteristic for the structures defined. Through statistical studies, we find the variability in mobility and sheet resistance as a function of the flake density.

(8)

(9)

Sommario

In un mondo in rapida evoluzione come quello dei semiconduttori, é tangibile il timore di stare per raggiungere limiti fisici nella continua riduzione delle dimensioni dei com-ponenti elettronici. I material bidimensionali (2D) sono stati selezionati come possi-bili candidati per essere sfruttati come canale all’interno di transistori ultra-scalati: il loro spessore atomico consente infatti un ottimo controllo elettrostatico da parte del gate, riducendo i cosiddetti effetti di canale corto, deleteri per il funzionamento di questi dispositivi. Diverse problematiche devono però essere affrontate affinché i materiali 2D possano essere integrati nelle tecnologie odierne. Tra questi lo sviluppo di metodi di fabbricazione in larga scala e la necessità di avere un supporto teorico per comprendere edi prevedere le caratteristiche elettroniche di nuovi materiali in modo da studiarne il comportamento all’interno di strutture innovative. In questa tesi sono stati sfruttati metodi di simulazione su multi-scala per valutare le performance di dispositivi elettronici realizzati con materiali bidimensionali in modo da guidarne la realizzazione. Questi metodi infatti permettono di includere vari livelli di astrazione e consentono una grande flessibilità così da assicurare un buon compromesso tra ac-curatezza e tempo di calcolo. In particolare, abbiamo in un primo caso analizzato le prestazioni di un transistore a effetto di campo (FET) realizzato sperimentalmente con un canale di MoS2 e una porta (gate) cilindrica di 1-nm di diametro sia

metal-lica che costituita da un nanotubo metallico, verificando che esistono interazioni di tipo source-drain ed effetti di tipo tunnel che riducono la resa in caso di canali piú corti di 20 nm. In secondo luogo ci siamo occupati di definire delle eterostruttura composta da "Noble-TMDs" di diverso spessore. Poiché con esso variano le proprietà elettroniche di questi, è stato possibile simulare il trasporto elettronico per strutture di uno stesso con regioni metalliche o semiconduttrici dipendentemente dallo spes-sore. È stato in questo modo possibile osservare valori di pendenza sotto-soglia minori di 60 mV/decade per dispositivi di PdS2 dovuti a meccanismi di "energy-filtering"

causati dalla particolare densità di stati energetici nel source. Sfruttando invece un’ eterostruttura di PtS2 é stato realizzato un diodo di tipo RTD (resonant-tunneling

diode) con un alto rapporto di corrente peak-to-valley, sfruttabile sperimentalmente. In ultima istanza abbiamo definito un algoritmo per la modellizzazione di strut-ture costituite da diversi flakes bidimensionali e la relativa simulazione di trasporto elettronico basato sull’ equazione di drift-diffusion e sull’estrazione di parametri di mobilita’ attraverso metodi ab initio.

(10)

(11)

List of contributions

This thesis is based on the published contributions listed below.

[1] E.G. Marin, M. Perucchini, D. Marian, G. Iannaccone and G. Fiori. Modeling of Electron Devices Based on 2-D Materials. In: IEEE Transaction of Electron Devices, vol. 65, no. 10 (2018), p. 4167-4179

[2] M. Perucchini, E.G. Marin, D. Marian, G. Iannaccone and G. Fiori. Physical insights into the operation of a 1-nm gate length transistor based on MoS2 with metallic carbon

nanotube gate. In: Applied Physics Letters vol. 113, no. 18 (2018), p. 183507

[3] E.G. Marin, D. Marian, M. Perucchini, G. Fiori, and G. Iannaccone. Lateral Het-erostructure Field-Effect Transistors Based on Two-Dimensional Material Stacks with Varying Thickness and Energy Filtering Source. In: ACS Nano vol. 14, no. 2 (2020), p. 1982-1989

Other unpublished articles by the author:

[A] Marta Perucchini, Damiano Marian, Enrique G. Marin, Teresa Cusati, Giuseppe Ian-naccone, and Gianluca Fiori Electronic transport in 2D-based printed FETs from a multiscale perspective. Submitted.

Comments on my own contribution

All the work presented in the papers listed above has been performed in close collabo-ration with the co-authors. A brief description of my personal contribution to the works is summarized hereafter.

- In paper [1] I took care of organizing the structure, preparing figures, I partly revised the literature and wrote the manuscript.

- In work [2] I performed the simulations, analyzed the results, prepared the figures and wrote the manuscript

- In paper [3] I performed DFT and Wannier preliminary calculations for PtS2as well as

transport calculations for the RTD device. I prepared part of the figures, participated in the analysis of the results and writing of the paper.

- In manuscript [A] I took care of preparing the new algorithm, performed the transport calculations (I did not perform DFT calculations), analyzed the results, prepared the figures and wrote the manuscript

(12)

(13)

CHAPTER

1

Introduction

I have not failed, I have just found ten thousand ways that will not work

Thomas Edison

Thomas Edison went down in history not only for discovering the carbon filament which changed completely electric lighting, but also for the extraordinary persever-ance, which has led to his invention. The legend says in fact that success came only after several attempts and failures. Even though the intention is surely laudable, the trial and error approach is a long-winded way of making discoveries but nevertheless very common, especially in material science [4]. In today’s fast paced semiconductor industry and economic scenario, finding the right pathway for progress among the zillions of possible wrong ones is a challenging task, paramount for being competitive and at the cutting edge of technology. This is especially true when it comes to a new and continuously expanding field, such as the one of two-dimensional (2D) materials. Until 2004 it was commonly believed that layered materials could not be isolated in a stable freestanding form but they would most likely degenerate at finite temperature [5, 6]. Everything changed however when in 2004 the first flakes of graphene were prepared and studied [7]. The promising behavior and electrical transport properties of this sheet made of carbon atoms sparkled vivid interest over the whole class. Since that moment several tens of 2D materials have been synthesized [8], and hundreds of alternatives to increase the list have been theoretically proposed [9–11] and the list increases nearly on a daily basis.

In Chapter 2 of this thesis, a selection of the most relevant materials of the 2D fam-ily will be shortly presented. This includes transition metal dichalcogenides (TMDs), the group IV X-enes, i.e. silicene, germanene and stanene, group III- and group IV-monochalcogenides, phosphorene and group-V layered materials, hexagonal-BN and layered oxides, etc., showing very diverse crystal structures (Figure 1.1-top). An extraordinary added value of layered materials is that their structure offer the

(18)

possibility of combining them in a virtually infinite number of compound configu-rations, also hybrid in dimensionality. A few examples of new artificial materials [12] built from these 2D blocks are reported in figure 1.1-bottom. Possibilities range from cutting them into quasi-1D nanoribbons, to stacking them in 3D van-der-Waals heterostructures [13]. Heterostructures can also be fabricated laterally from different phases of the same material [14, 15] or with different number of stacked layers [16] resulting in interfaces with different dimensionality. The creation of networks of ran-domly positioned nanotubes [17] or flakes [18] with a possible application in flexible and wearable electronics are a next step in the adventure [19].

A part from a fundamental research point of view, 2D materials have sparkled substantial interest because they can be exploited in a wide range of possible applica-tions in electronics [22, 23], photonic, optoelectronics and even valleytronics thanks to their very intriguing physics, so different from their bulk counterpart. This includes superconductivity, topological properties, spin transport and magnetism, phase tran-sitions, or charge density waves [24, 25] but a more "evident" property has been endorsed to unravel its full potential in electronics applications: the atomic thick-ness. In fact at the heart of field-effect transistors is the control of the potential energy of the electron in the channel which can be quantified in terms of the scaling length λ = ptstox(s/ox) (where ts and s are thickness and the dielectric

permittiv-ity of the body, respectively, while tox and ox and the thickness and the dielectric

permittivity of the gate oxide, respectively [26]). Qualitatively speaking, the smaller λ, the greater the control of the gate over the channel and it comes straightforward how the atomic thickness of 2D materials guarantees the minimum ts achievable.

Moreover, the atomic smoothness of the layered 2D materials reduces scattering phe-nomena, which is beneficial for another quantity of interest in electronics: mobility [27]. These and many other properties can be combined in order to apply 2D materi-als to electronics and to new device, which are expected to be in the limelight of the electronics research community in the short- and mid-future.

However, in spite of this promising potential for electronics and the great advances in the last decade, a major drawback in the preliminary investigation and in future industrial integration of 2D materials is related to their production and processing techniques. These are not mature enough, although progresses in synthesis of wafer-scale graphene [28] and TMDs [29] have been made. For this reason is even more important to convey the experimental efforts only on those materials and architectures showing the best expected performance, since testing every new theoretical alternative becomes prohibitive. Simulation and modeling are, in this context, indispensable tools to perform timely and low-cost exploration and prediction. Considering the complexity of current nanoelectronics and the peculiar specific 2D physics observed in such materials, the theoretical approach requires versatility so to adapt and capture on the one hand the details of the materials behavior and at the same time to provide a high-level description of device operation.

For a proper study of 2D-materials-based electronic devices indeed, these descrip-tions must be combined in a flexible approach where more accurate results are used at

(19)

Chapter 1. Introduction 3

Figure 1.1 | Top: Lateral and top views of the crystal structures present in some 2D materials, e.g. TMDs (left) III-chalcogenides, group IV and black

phosphorus.

Bottom panel: structures of diverse dimensionality built from these 2D blocks, including 2D lateral heterostructures, 3D vertical or van-der-Waals heterostructures (top left), and networks (bottom left) [Reprinted from [20,

(20)

higher abstraction levels, and materials properties feed device modeling through well-defined interfaces for exchange of information (as it will be discussed in 3). This is the purpose of the multi-scale method that has become a standard and a increasingly adopted approach in the 2D materials community [30], thanks to the possibility of choosing the best trade-off between accuracy and computational demand. A review on this subject by the author and colleagues has been published in [1].

In the following chapters we take advantage of multi-scale simulation techniques to provide insights for the fabrication of new 2D-material based device concepts. The mechanisms underlying electronic transport are investigated in order to provide a framework for the potential performance of the transistors and their applications. The results presented in this thesis have also been published in Ref. [2] and [31].

We first examine in Chapter 4 a FET with a monolayer MoS2 channel, controlled

by a cylindrical gate with the diameter of 1 nm. Shrinking the gate length has always been a target since it is considered a defining dimension for the transistors. A uniform scaling is however required for integrated circuit components and for this reason we study the effect of channel length scaling on the output current for this type of device. Moreover, we compare the results with those obtained for a semi-metallic carbon nanotube gate of the same dimensions, a configuration already experimentally proven in the work of Desai et al. [32], evaluating the effect of the finite density of states of a CNT gate on the loss of device performance (see. [2]).

In Chapter 5 instead we deal with a different device architecture, a lateral het-erostructure, proposed to deal with another issue concerning the realization of 2D material based FET, namely contact resistance. By exploiting the bandgap depen-dence of the so-called noble transition-metal dichalcogenides upon the stacking layer we juxtapose source, channel and drain regions using a single material with different thickness, avoiding abrupt changes at the interface and thus minimizing contact re-sistance. We also propose a resonant tunneling diode taking advantage of the same phenomena (see. [31]).

Finally, in Chapter 6 we apply the multi-scale simulation methodology to study devices composed by networks of 2D material flakes such as graphene or MoS2,

re-alized by inkjet printing. This manufacture technology in fact allows cost-effective, rapid and versatile definition of device components but a theoretical framework to understand the properties of these heterogeneous structures is still lacking. We thus define an algorithm for studying electronic transport in such FETs and evaluate these results with respect to factors which can be linked to process parameters in order to optimize them.

(21)

CHAPTER

2

2D materials and 3D structures

2.1 Introduction and chapter outline

The scientific community has undoubtedly chosen the year of the successful isola-tion of the first graphene flake by Novoselov et al. [7], 2004, as the birth of two-dimensional (2D) materials research field. From that moment on, in fact, researchers have explored, both on the computational and experimental sides, the possibility to obtain other stable layered materials. After almost two decades, it has been shown that many exist already and even more could potentially, spanning all the range from metals to insulators, including of course the semiconductors [33] (a few of them are reported in figure 2.1 (a)).

In this chapter, we will briefly summarize the key features and properties of these materials and a particular focus will be dedicated to graphene and transition metal dichalcogenides (TMDs) given their relevance to this work. In section 2.3, instead, we will see the main device structures realized with these innovative materials.

2.2 2D layered materials

2.2.1 Graphene and other X-enes

As already mentioned, graphene with its intriguing physics and successful isolation in 2004 [7] surely deserves the credits for opening the pathway to the investigation on two-dimensional materials. Most of its particular behavior resides in its monolayer honeycomb planar structure consisting of bonded sp2 _{hybridized carbon atoms and}

half-filled p orbitals, perpendicular to the plane and partially overlapping. These delocalized electrons are exactly those who determine the transport properties of graphene together with its band structure. In fact, the dispersion relation of graphene (Eq. 2.1) is linear for small enough energies and cannot be well approximated by a

(22)

parabolic band but instead takes the shape of what is called "Dirac cones", touching at the K and K’ points of the first Brillouin zone.

E(k) = s~vF|k| with |k| =

q k2

y + k2z (2.1)

where vF ∼ 108 cm/s is the Fermi velocity and s = ±1 for CB and VB,

re-spectively, ky and kz the in-plane wavevectors. As it becomes clear, graphene is a

semimetal, lacking a proper bandgap (which can, however, be opened in the bi-layer configuration or by restricting the lateral size as in the case of nanoribbons) but its many other properties can be exploited in electronic applications such as high room-temperature carrier mobility [34], high mechanical tensile strength (130 GPa) [35] and thermal conductivity [36], and lastly an almost perfect transparency [37]. Other single-material hexagonal lattices have drowned attention such as the buckled phosphorene (black phosphorus (BP) monolayer) [38] with its thickness dependent, sizable bandgap appealing for optoelectronics applications. Besides, to the X-ene family belong also the gapless silicene, stanene [39] and germanene [40], the graphene counterpart for Si [41, 42], Sn and Ge but although these materials may represent a good opportunity for low-power devices many technological problems need to be over-come for their application, from thermodynamic stability to understanding substrate interaction [43]. Very recently also bismuthene, antimonene and their compounds have been the object of investigation mainly thanks to their spin-orbit coupling and potential topological insulator properties [44, 45].

2.2.2 Transition-metal Dichalcogenides (TMDs)

Transition metal dichalcogenides are often referred to as MX2 in reference to their

monolayer composition: one layer of metal atoms (M=W, Mo, Nb, etc.) sandwiched between two chalcogens (X=S, Se, Te) layers, disposed on a hexagonal lattice. Two polymorphs are possible (as represented in figure 2.1 (a)) depending on the metal coordination: trigonal prismatic and octahedral which together with stacking give rise to three different polytypes [46](see figure 2.1 (b)). The metastable 1T structure is typical of the TMDs metal phase and has the metal in the octahedral configuration whereas the more stable 2H structure (semiconducting phase) consists of the repeti-tion of AB stacked layers in which the metal has a trigonal prismatic configurarepeti-tion [47]. What makes semiconducting TMDs the actual possible "substitutes" of Si is that they possess a bandgap in the range of 1 eV and more precisely its magnitude increases with the thinning of the material [48], which in some cases, like that of MoS2, also leads to a shift from indirect to direct bandgap. The calculated values go

from 1.71 eV for monolayer 2H-MoS2 to 0.88 in bulk form, whereas experimentally

the single-layer gap reported was 2.16 eV [49]. These features are appealing both for electronics and optoelectronics applications, supported by other TMDs properties such as high mechanical strength (almost 1 TPa for in-plane stress on single-layer MoS2) and thus resistance to bending [50] together with reasonable mobility values

(23)

2.3. Devices and structures made of 2D materials 7 going from a few to 500 cm2_{/Vs depending on the material and substrate quality}

[51–53].

2.2.3 Hexagonal Boron Nitride

Two-dimensional h-BN stands out in the 2D material being by itself an electrical insulator. Its lattice is composed of alternating B and N atoms in a honeycomb-like structure, analog to that of graphene but because of the high asymmetry of the B and N sub-lattices, the resulting material band gap is very large (5-6 eV). In addition to its insulating properties, h-BN has demonstrated to enhance mobility of other 2D materials (graphene, TMDs, etc.) when used jointly, by protecting them from contamination, oxidation and thermally/electrically induced degradation [56, 57].

2.2.4 Other 2D materials and compounds

Other material compounds involving chalcogens are the so-called SMCs in which S or Se are combined with a semimetal (M=In, Ga) in M2X2 four-layer configuration

[33, 58–60]. InSe, for instance, can be interesting for its ambient stability and the bandgap matching the visible region. The list of potentially interesting bi-dimensional materials could literally continue endlessly, going from 2D material metal oxides [61], still under preliminary investigation, to graphane, whose first prediction [62] and experimental [63] demonstration date back to 2007 and 2009, respectively. However, they will not be discussed hereafter but what appears evident is how fundamental theoretical research and practical realization and application of these materials need to go hand-in-hand to focus the effort on achievable and worthy paths.

2.3 Devices and structures made of 2D materials

The observation made by Gordon Moore in 1965 that the number of components per integrated circuit would double every 18 months [64] has been questioned fre-quently in recent years because of the challenges and difficulties in keeping up with the expectations. What is now known as Moore’s law has, in fact, become a defini-tion for regular innovadefini-tion in processor manufacturing, assiduously followed by the tech industry. This has relied on the scaling of transistors thus on the reduction of components minimum feature size, such as the gate or channel length in traditional MOSFET (Metal-Oxide-Semiconductor Field-Effect-Transistor). The advances in Si-based technology have been promoted by continuous improvement of the processing techniques including lithography, thin film deposition and surface treatments as well as by the introduction of high-κ oxides, stress engineering and tri-gate geometries [23, 65]. Approaching atomic thickness with silicon however implies several draw-backs such as the degradation of mobility, increase in surface roughness and loss of gate control over the channel which results in an overall performance variability. It is here, below the 10 nm node that 2D materials with their intrinsically thin and

(24)

a)

b)

c)

Figure 2.1 | (a) Metal coordination types for TMDs: octahedral (top) and trigonal prismatic (bottom); (b) Top and lateral view of 1H, 2T and 3R poli-types; (c) Crystalline structure representation and band structure for various

(25)

2.3. Devices and structures made of 2D materials 9 uniform structure come into play [66–68]. They, in fact, offer distinct advantages such as a surface free of dangling bonds, a lower mobility degradation with thickness reduction and stability under strain which can all be exploited in the various device architectures whose recent fabrication advances will be described in the following sections.

2.3.1 Lateral FETs

Field-effect transistors (FET) offer a simple yet fundamental way to highlight mate-rial electrical properties. In these devices, current flows between two metal contacts (source and drain) through a channel (usually a semiconducting material), whose conductivity is modulated through the voltage applied at the gate, the third con-tact. It controls the passage of charge carriers by raising or lowering the potential barrier for carrier thermal energy. The "ON" and the "OFF" states for the device thus correspond to highly conductive (low barrier) or resistive (high barrier) channel conditions, respectively as in Fig.2.2(left). The extremely high carrier mobility es-timated for graphene (above 2 × 105 cm2_/Vs) [69] has surely aroused interest in its

application as channel material in transistors. In the works of [65, 70, 71] the mobil-ity values reported for monolayer graphene FETs (GFETs) at room temperature vary from 1.8 × 104 _cm2_/_{Vs for a SiO}

2-Si back-gated to 7.0 × 105 cm2/Vs for metal/HfO2

top-gated structure and finally to 1.4×105 _cm2_/_{Vs for an h-BN/graphene/h-BN}

het-erostructure with an edge-contact geometry. h-BN has been shown to be a better substrate for graphene —which is very sensitive to its environment—for multiple rea-sons [72, 73] among which the reduction of impurity scattering that leads to overall better channel µ. Yet, the highest mobility value remains that of suspended graphene [34, 74] reaching peak values of 230000 cm2_/_{Vs. Mobility, however, is not the only}

important parameter for a transistor functioning and graphene lack of band-gap pre-vents it to be used in logic application since it results in very low on-off current ratios (ION/IOFF) ≈ 1-10. Besides, high sub-threshold swing (SS), which represents

the increase in gate voltage required to obtain a gain of one order of magnitude in the current, are also detrimental characteristics for GFETs. On the contrary, devices based on 2D channel materials with sizable energy bandgap (around 1 eV) such as TMDs exhibited ION/IOFF of several orders of magnitudes although at the expenses

of a lower µ compared to that of GFETs. The first estimation of MoS2 mobility

from fabricated FETs reported values of 0.5 to3 cm2_/Vs [75], too low for practical

applications. Nonetheless, soon after various techniques were developed to enhance transistor performance with a scaling perspective such as in [66, 76–79] where MoS2

mobility touched the value of 700 cm2_/_{Vs. A lot of effort has gone into the}

re-duction of contact resistance using non-conventional metals such as scandium [80], graphene [81] or substoichiometric-molybdenum trioxide [82] electrodes or by doping [83] but the issue remains an open question to be addressed before these materials can be competitive with existing technologies [84, 85]. Similar challenges extend to other TMD materials such as WSe2, the p-type counterpart of MoS2 which exhibited

(26)

Figure 2.2 | Operating mechanisms for thermionic FET in the OFF and ON states (a) and I-V curve (b) together with a graphical representation and

experimental realization of a MoS2 based device (c), adapted from [92].

on/off-current ratio of 107 _{and holes µ of approximately 300 cm}2_/_{Vs in the work of}

[86], ReS2 [87] or HfS2 [88]. Together with contact resistance, large scale fabrication

still represents a confronting question together with related problems like the effect of grain boundaries and crystalline quality [89–91]

2.3.2 Vertical heterostructures, tunnel FETs

Given the difficulty to achieve very low off-currents (fundamental for low power con-sumption application) with lateral graphene or TMDs based FETs, combinations of different 2D materials vertically stacked were proposed. A typical operating mech-anism of this type of transistor is based on the tunneling effect (so they take the name of TFETs): the possibility of injecting charge carriers in the channel relies in fact on the availability of allowed states there rather than on carriers thermal energy, as in thermionic devices allowing to reach sub-threshold swing values lower than 60 mV/decade. More specifically, in the ON state, band-to-band tunneling is possible since the alignment of the bands controlled by the gate (visible in Fig. 2.3(a)) results in a thin barrier; conversely, in the OFF condition there are no states available in the channel and tunneling phenomena are suppressed. A graphene/h-BN/graphene TFET working has been realized by Britnell et al. [93] however with very low on-current density. With this regard, graphene/MoS2 heterojunctions have

demonstrated on/off-current ratios higher than 103 as in [94–96]. TFETs with high

ION/IOFF have also been realized exploiting graphene in combination with other TMD

layers such as WS2 [97] or in a WSe2/SnSe2 van der Waals vertical heterojunction

[98]. New transport mechanisms have also been highlighted as in the case of the graphene/WSe2 vertical device proposed by Shim et al., for which the tunneling

phe-nomena is originated by WSe2 defect states aligned near the graphene Dirac point

[99]. Finally, in vertical structures tunneling and thermionic currents can be coop-erating processes, as shown for the graphene and black phosphorus structure in the work of [100].

(27)

2.3. Devices and structures made of 2D materials 11

Figure 2.3 | Operating mechanisms for a TFET in the OFF and ON states (a) and I-V curve (b) together with a graphical representation and experimen-tal realization of the ATLAS-FET based on Ge-MoS2, adapted from [101].

2.3.3 NDR devices

Negative differential resistance devices (NDR) are identified by non-monotonic IV characteristics, which conversely show n-shaped curves providing multiple threshold voltage values. Depending on the underlying functioning principle, they can be di-vided into Esaki diodes or RTD (Resonant tunneling diodes)[102]. The former works by interband tunneling (as schematically depicted in figure 2.4): with forward bias, the electrons in the conduction band in an n-type semiconductor flow into the empty states of the valence band in the p-type region. When the applied voltage pushes CB above VB, electrons can no longer tunnel and we assist at a reduction in the current which will again increase at the start of thermionic processes. 2D-material based Esaki diodes working at room temperature have been demonstrated for BP-SnSe2 [103] and BP-ReS2[104] heterostructures, whereas MoS2-WSe2heterojunctions

showed NDR at low temperatures [105]. Differently from Esaki diodes, RTDs are based on intra-band tunneling between two potential barriers, among which resonant states exist with an energy ER. When a voltage is applied, while the resonant states

are pulled down to align with the left contact (emitter), the current increases reaching a maximum value to decrease again when ER falls below the conduction band of the

emitter. In this condition, electrons can no longer tunnel and the current must flow through scattering channels or by thermionic emission [102]. It has been shown that vertical stacking of 2D materials results in a natural formation of a double potential barrier, without constraint such as lattice matching of the layers. RTDs have been proposed [106] and demonstrated mostly with h-BN sandwiched between mono- or bilayer graphene, for which the crystallographic orientation plays a fundamental role [107–109]. Moreover, multiple resonant peaks have been observed in correlation with the alignment of the two sub-bands of the two bilayer graphene [110]. NDR behavior has also been investigated and witnessed in TMD based devices as in the case of three-layer MoS2 in between nanoporous electrodes at low temperatures [111] or f

(28)

Figure 2.4 | Operating mechanisms for an Esaki diode at different applied voltages (a) and I-V curves (b); (c) schematic representation and optical image

(29)

CHAPTER

3

Modeling methods for 2D materials

3.1 Introduction and chapter outline

Considered the vast panorama of devices architectures and materials that 2D material based electronics can draw from, it is essential to have a similarly extensive pool of modeling techniques that can adapt to single cases to understand and predict electronic transport in these structures. In this Chapter, the most commonly used methods will be revised, subdivided into two main categories: the first is dedicated to the physical properties of the 2D material under study, which mainly consists in the bandstructure (see paragraph 3.2); the second instead takes care of the tools required to analyze the behavior of the device when different voltages are applied to its terminals (in paragraph 3.3). In doing so, the modeling techniques used in the works proposed afterward (in Chapter 4, 5 and 6) will be highlighted (see figure 3.1). In particular, the focus will be on the combination of various methods in a multi-scale modeling approach.

3.2 Materials modeling

All the physics that takes place in a device is determined by the constituting materials, specifically by the precise arrangement of the atoms and their interactions. For an electron device, the main information on the materials is embedded in the electronic band structure, i.e. the energetic representation of the allowed electronic states, that provides most of the electronic variables of interest, e.g band-gap, electron affinities, work function, density of states, effective masses, etc.

The exact calculation of the electronic band structure of materials requires to solve the interacting many-particle Schrödinger equation, which is unfortunately too complicated to be solved except for extremely simple cases. However, there is a complete hierarchy of methods to approach its solution (Figure 3.1).

At the base of this hierarchy, there are ab-initio or first principle methods, in-cluding Quantum Monte Carlo, Density Functional Theory, and GW approximation.

(30)

Figure 3.1 | Scheme of the different levels of approximation for the model-ing of materials electronic properties hierarchically ranked accordmodel-ing to their accuracy and computational demand (left); scheme of different approaches for device modeling, hierarchically ranked (right). The multi-scale methods used to study the devices presented in the thesis are highlighted with different

(31)

3.2. Materials modeling 15 Atomistic models constitute the next level of approximation: they are quantum me-chanical in form, but based on fitting parameters that are extracted from ab-initio or empirical results and include Tight-Binding, Maximally Localized Wannier Functions or the k·p approximation. The rank in the material modeling ends up with analyti-cal approximations of the electronic band-structure. In the following, we will briefly review these methods and their application to the investigation of 2D materials.

3.2.1 Quantum Monte Carlo (QMC)

The QMC methods allow the solution of the 3N-dimensional Schrödinger equation by handling the explicit many-body formulation of the problem thus considering a great number of interacting electrons [113]. The essence of this method relies on the evaluation of a multi-dimensional integral by statistically sampling the integrand and averaging the sampled values. Amongst the different QMC methods, the most popular one for the calculation of the electronic band structure are the variational Monte Carlo (VMC) and the diffusion Monte Carlo (DMC) [114]. Nowadays the QMC techniques in the framework of the 2D material are mainly used as a benchmark for testing Density Functional Theory results. Indeed, as they are free of approximations (except for their dependence on the initial trial wave function) these techniques are superior for studying layered systems where van-der-Waals interactions play a crucial role [115–117], but their extremely high computational cost limits their use.

3.2.2 Density Functional Theory

(DFT) is the most widely used method to determine the electronic properties of solid-state systems. Despite well-known inaccuracy in determining semiconductors bandgap, it is the touchstone for most of band structure calculations. Instead of solving the many-body problem, it takes advantage of the fact that the properties of an interacting electron system can be as well described by a functional of its ground state density. The Kohn and Hohenberg theorems[118] establish the mathematical foundation of DFT [119]. The first asserts a one-to-one relation between the external potential and the ground charge density for a system with a given number of particles, albeit not specifying the form of the function that relates them. The second theorem, instead, says that "the ground state energy of the system at any particular exter-nal potential is the global minimum of the energy functioexter-nal and the density which minimizes the functional is the exact ground state density" [118, 119]. These two the-orems are translated into practice by the Kohn and Sham approach [120] where the many-electron interacting problem is substituted by a many-electron non-interacting problem with identical ground state charge density. The energy functional is then written as: EKS[n] = Ts[n] + Z d3r Vext(~r) n(~r) + 1 2 Z d3 d3r0 n(~r) n(~r 0₎ |~r − ~r0_| + Exc[n] (3.1)

(32)

where Ts[n] represents the kinetic energy for a non-interacting electron system, the

second term specifies the contribution of the external potential (due to the nuclei and other external potentials) and finally the third term, called the Hartree term, is the classical electron-electron Coulomb interaction contribution. Exc is known as the

exchange and correlation functional and absorbs the differences between interacting and non-interacting many-body systems. Vxc is its related potential that, if known,

would make the Kohn-Sham approach an exact solution of the many-body interacting stationary Schrödinger equation. However, this is so far not possible and the Vxc must

be approximated. Accordingly, the accuracy of the DFT calculations depends directly on the choice of the approximation. Among all the possibilities, the most commonly used are the Local Density Approximation (LDA) and the Generalized Gradient Approximation (GGA). The first, proposed directly by Kohn and Sham [120], exploits the simplest form of Exc, which is locally that of a homogeneous electron gas. The

GGA refines the LDA by including the dependence of the exchange-correlation energy not only on the charge density but also on its gradient, as Hohenberg and Kohn proposed in reference [118]. To extend the reasoning to solids, periodic boundary conditions need to be accounted for. These, together with the Bloch theorem [121], allow reducing the problem of an infinite number of electrons to an infinite number of k-points, which by proper sampling can be used to describe the whole solid electronic structure sufficiently well. Another factor influencing the accuracy of DFT based results is the choice of a proper basis set for the description of the wave functions in the whole region. Some examples are a linear combination of atomic orbitals (LCAO) or the plane waves (PW) basis set. The latter has been extensively used since it does not depend on the system gives a good performance in terms of convergence. The application of DFT methods to the study of 2D materials forms an extensive literature discussing graphene, transition metal dichalcogenides [122], group IV [123] and group V [124] layered materials, III-V layered compounds [125] and practically any 2D material conceived so far [9].

3.2.3 GW

GW reformulates the many-particle Schrödinger equation, under the Green’s func-tion theory, into a set of single-particle-Schrödinger-like equafunc-tions where exchange and correlation effects are included in the so-called self-energies, Σ = iGW , that give the name to the approximation. The set of equations is solved iteratively and self-consistently usually starting from a tentative solution that is taken from a converged DFT simulation. The GW method goes beyond DFT in considering the electron-electron interactions employing exact, nonlocal Fock exchange and correlation dy-namical screening, thus giving a better estimation of the bandgap of semiconductors with respect to DFT, but at the cost of higher computational demand. To this purpose, simplified implementations as G0W0, GW0 that relax the self-consistency

requirements are often employed. GW together with DFT has been used to accu-rately determine the band structure and effective masses of a wide collection of 2D

(33)

3.2. Materials modeling 17 materials in [126], which can be later used in device-level abstractions.

3.2.4 Tight-binding

The tight-binding (TB) method is one of the most used methods in the device commu-nity due to its computational efficiency and accuracy. It can be formulated starting with a linear combination of atomic orbitals of Bloch-form localized at each site of the atomic lattice. The tight-binding ansatz wavefunction is written (for one orbital per lattice site) as: |Ψi = Pme

ik·Rm_|mi/√_N, where |mi is the state at site m, R

m

is m site lattice vector, and k is the wavevector. The energy band structure can then be obtained by plugging |Ψi into the Schroedinger equation and projecting it into hΨ| obtaining: E(k) = P n,me ik·(Rm−Rn)_{hn| ˆ}_H|mi P n,meik·(Rm−Rn)hn|mi

where hn| ˆH|miand hn|mi are used as input parameters obtained empirically or from first principle methods. Several tight-binding models have been developed for 2D materials, making it the most used approach for band structure calculations amongst Atomistic methods. It has been employed to study e.g. the band structure of TMDs [127, 128], black phosphorus [129], or the effect of the spin-orbit coupling on silicene, germanene and stanene [130], the impact of strain on graphene [131], or the chirality in carbon nanotubes [132]. TB models can capture the lower energy band structure, but usually have some difficulties in reproducing high energy states with a limited number of nearest neighbors. Its extension to higher unoccupied energies in 2D materials has been treated in [133].

3.2.5 Maximally Localized Wannier Functions (MLWF)

The MLWF method represents the ideal connection between ab-initio calculations and tight-binding Hamiltonians. Wannier Functions [134] retains the information of Bloch states, ψnk, in a single-particle approximation like DFT, such that, the

electronic ground state of the system is mapped onto localized Wannier states, wnR.

The mapping of the Bloch states into Wannier functions is not unique, since there is Gauge freedom [135], with the subsequent implication that from a set of Bloch states one can have different sets of Wannier functions. MLWF represents among these different possibilities the most localized ones. The procedure of seeking the most localized Wannier functions was firstly described for an isolated group of bands [136] and later extended for and an entangled group of bands [137] and implemented in the code Wannier90 [138]. The usefulness of passing from extended states to localized states resides on the possibility of deriving a tight-binding-like Hamiltonian from the Wannier functions in a fully automatic fashion, skipping time-consuming procedures to obtain an optimized set of parameters as in the semi-empirical tight-binding methods, through the least-mean square or genetic algorithms. The Wannier Hamiltonian has been successfully adopted in the study of 2D materials to later

(34)

compute transport properties. Only to name a few, MLWFs have been used to confirm the presence of edge states in 2D bismuthene alloys [139] calculate transport in arsenene and antimonene 2D FET transistors [140], to determine the Landau levels and the corresponding edge states of several 2D crystals including graphene, hBN, MoS2, phosphorene and InSe, [141], to evaluate the performances of 2D FET based

on InSe [60], phosphorene and MoS2 devices [142–144], to analyze the conductance

of functionalized carbon nanotubes [145], to study two dimensional transistor based on heterostructures of MoS2 1T and 2H phases [47] and to propose a TFET device

based on stanene nanoribbon [146].

3.2.6 k·p

k·p method is a semi-empirical model based on perturbation theory that allows the extrapolation of the bandstructure over the Brillouin zone once the description of the zone center is known (i.e. the bandgap and the optical matrix element). Although k·p must be fed by input parameters from ab-initio or tight-binding models, it provides a simple and accurate enough depiction of the electronic properties at low energies. A recent review presents an extensive collection of k·p parameters for transition metal dichalcogenides [147]. The k·p methods have also been employed to model silicene and germanene band structures at low energies [130].

3.3 Device modeling

To model the behavior of an electronic device, it is necessary to determine the carrier distribution out of the equilibrium, i.e. when some external voltages are applied at its terminals. Differently from materials modeling, where a quantum approach for the electronic band structure is mandatory, the length scales involved in device modeling enable one to choose between a quantum description centered on the Schrödinger equation and a semi-classical approach based on the Boltzmann Transport Equa-tion (Figure 3.1). Consequently, the hierarchy of methods includes i) quantum-based methods like the conditional wave function (CWF) or Non-Equilibrium Green Func-tions (NEGF) and ii) semi-classical descripFunc-tions based on the solution of the Boltz-mann Transport Equation (BTE), e.g. Monte Carlo transport, Drift-Diffusion (DD). At the bottom of the hierarchy, simplified analytical models appear. There is a vari-ety of ways in which the materials description information enters in these methods, from Tight-Binding depictions in NEGF, to definitions of the dispersion relationship at low energies for CW, or the density of states for DD. Nevertheless, for a consis-tent solution, the study of the device must include the electrical connection between the external potential and the charge carriers through the Poisson equation. In the following, we review the aforementioned methods to calculate transport in electronic devices and some of the main studies employing them in 2D based transistors.

(35)

3.3. Device modeling 19

3.3.1 Non Equilibrium Green Functions

The NEGF solves the open-boundary Schrödinger equation [148]. It determines the density matrix of a device connected to two or more contacts, characterized by dif-ferent Fermi levels, taking into account correlation effects on the energy states occu-pation. All the information required to study transport in the device is contained in the Green’s function, that gives the response of a system to a constant perturbation in the Schrödinger equation [149]. In fact, the Green’s function is much easier to calculate than solving the whole eigenvalue problem and is defined as:

(ES − H) G(E) = I (3.2)

where S is the overlap matrix, H is the Hamiltonian matrix and I is the identity matrix. If we divide the system into three regions, namely the left (L) and right (R) contacts and the central device (D) we can rewrite H and G as

H =   HL τL 0 τ_L† HD τ † R 0 τR HR   G =   GL GDL GRL GLD GD GRD GLR GDR GR   (3.3)

where τL/R stands for the interaction between L/R and the scattering region D

and HL, HD and HR denote the Hamiltonian matrices of the respective regions. By

solving equation 3.2 with the expressions in Eq. 3.3, GD is given by:

GD = [ES − HD− ΣL(E) − ΣR(E)]−1 (3.4)

where ΣL/R(E) = τL/R† gL/R τL/R are the so-called self-energies, accounting for the

contribution of all the semi-infinite leads attached to the device. An additional term (Σph) can be included to account for inelastic scattering phenomena (as those from

electron-phonon interactions) in the device region. In this way, it is possible to reduce the calculation from the infinite Hamiltonian dimension to that of the sole central part. NEGF can provide important information as the free carrier concentration and the transmission coefficient (T ), from which the current can be calculated thanks to the Landauer formula [150]. T , without going into the details of the derivation, can indeed be written as

T (E) = T r(GD†ΓRGDΓL) (3.5)

with the energy level broadening term ΓL/R = −2Im[ΣL/R(E)]. When applying

the NEGF method to the study of devices, it is important to bear in mind the assumptions it is based upon, namely the single-particle approach and the mean-field approximation. Therefore, this formalism is not suitable in cases of strongly correlated transport or in the ‘Coulomb blockade’ regime. Nonetheless, it is a central tool in the determination of the transport in 2D-based electronic devices and has been widely employed to determine transport in a variety of 2D-based structures, including carbon nanotubes [151], TMDs transistors [144, 152–154] and heterostructures [47,

(36)

155], monochalcogenides FETs [60, 142, 156], black phosphorus and other group V layered materials transistors [140, 157, 158], or group IV nanoribbons [146, 159].

3.3.2 Conditional wave function (CWF)

The conditional wave function approach has been recently developed to study quan-tum transport in 2D (and 3D) materials. This method arises from Bohmian Mechan-ics, an alternative version of Quantum Mechanics in which particles have a definite position and their motion is choreographed by the wave function [160–163]. In partic-ular, the CWF is the wavefunction of a subsystem defined in this theory that serves to propose an alternative solution to the so-called many-body problem. The CWF of each particle evolves under a Schrödinger-like equation where Coulomb interactions and the external voltage are treated exactly, while two unknown potentials, which re-tain the non-local correlations of the many-body wavefunction, must be approximated [164]. This method has been recently applied to investigate quantum dissipations in two-dimensional materials with graphene-like band structure [165].

3.3.3 Monte Carlo method

The Monte Carlo (MC) statistical solution of BTE has been one of the most widely used methods for studying transport in traditional semiconductors [166]. It is con-sidered as a semi-classical approach since it takes advantage of statistical analysis to track the trajectory of a particle, whose motion is treated in a classical way as long it is stochastically scattered. The scattering phenomena are instead treated quantum-mechanically using Fermi’s golden rule [167, 168]. MC solution of BTE does not make any assumption about the distribution function and is suitable for high-field scenarios and among the variations of the MC approach, the ensemble method is con-sidered the most appropriate for nanoscale devices. MC has been used to investigate transport in several 2D based electronic devices as in [169–173].

3.3.4 Momentum Relaxation Time (MRT)

A simpler approach to the BTE is the Momentum Relaxation Time (MRT) approx-imation, which is valid for small displacements from equilibrium. It works well in determining transport parameters such as mobility and diffusion coefficients in uni-form conditions, thus where these quantities do not depend on the position [174]. The MRT approximation is commonly used in conjunction with simplified band-structures based on tight-binding or analytical approximations in the close vicinity of the band minima or maxima. Several works have made use of the MRT to calculate the scat-tering limited mobility in, e.g. graphene and graphene nanoribbons [175–177], MoS2

(37)

3.3.5 Drift-Diffusion

Drift-Diffusion (DD) has been one of the most used approximations to model trans-port in traditional semiconductors and the literature treatment is extensive [182, 183]. It can be derived from BTE under the relaxation time approximation, resulting in independent expressions for the electron/hole current:

Jn(p) = Jdrift+ Jdiff = q n(p)µn(p)∇V ± q∇n(p)Dn(p) (3.6)

where n, p, µ, and D are the electron and hole charge density, mobility and diffusion coefficient, respectively. The first term of the expression accounts for the drift velocity, which results from an applied electric field, with µn/p describing how

quickly carriers move through the lattice: it is a property of the material, inversely proportional to the effective mass of the charge carriers (m∗_{) and directly related to}

the mean free time between collisions (τ), which may depend on different underly-ing scatterunderly-ing mechanisms, whose effects are summed through the Matthiessen’s Rule [184]. The second term of equation 3.6 refers instead to the diffusion process, which is a consequence of a non-uniform distribution of charge carries so that a concen-tration gradient (∇n, ∇p) is present, as in the case of non-uniform doping profiles in semiconductors. Then, to reach thermal equilibrium, charges need to move from regions of higher to lower concentration obeying Fick’s first law as in Jdiff, where the

proportionality term D [cm2_s−1_{] is the diffusion coefficient, whose relationship with}

the mobility is given by Einstein’s relation: Dn(p)=

kBT

q µn(p) (3.7)

Both µ, and D can depend on the electric field and since both parts of the con-stitutive law (Eq. 3.6) are strictly related to the Fermi level (EF), the expression can

be reduced to a single term, function of EF1:

Jn= µnn · ∇EF Jp = µpp · ∇EF (3.9)

1_{In the assumption of non-degenerate semiconductors, the F ermi-Dirac distribution function}

can be well approximated by the M axwell-Boltzmann0s statistics so that the electrons and holes concentration can expressed as n = NC· exp

_E F−EC kBT and p = NV · exp _E V−EF kBT , where NC and

NV are the effective density of states, ECand EV the minimum of the conduction and the maximum

of the valence bands and kB the Boltzmann’s constant. Simplifying the discussion to the electrons

for the 1D case without loosing generality we can write the spatial derivative of n as: dn dx = NC kBT · exp EF− EC kBT · dEF dx − dEC dx = n kBT · dEF dx − q∇V (3.8) so that inserting Eq. 3.8 in Eq. 3.6 and substituting D as in Eq. 3.7, Jn becomes:

Jn = qnµn∇V + µnn ·

dEF

dx − µnnq∇V = µnn · dEF

(38)

This derivation of the drift-diffusion equation is rigorous for thermal equilibrium conditions when the product of n and p is equal to the square intrinsic carrier con-centration ni and the Fermi level is uniquely defined for both types of charge carriers

at each point of the device. When an external voltage is applied, however, the system transitions to out-of-equilibrium conditions, which means that the product of elec-tron and hole concentrations (np) moves away from n2

i to larger values in the case

of carriers generation or smaller values for carriers recombination. In this situation, although electrons and holes are at local equilibrium with themselves, they are no longer in equilibrium with each other and the Fermi level splits into two quasi-Fermi levels: EF,n and EF,p (also sometimes referred to as imref). In this scenario, the

quasi-Fermi levels hold the same physical meaning of EF. Lastly, it must be pointed

out that when recombination/generation times are very short (1-100 ps) EF,n and

EF,p cannot diverge substantially, thus considering a single quasi-Fermi level is a

valid approximation. [185–187]

DD needs to be solved in conjunction with the continuity equation that might introduce a time-dependent variation of the carrier densities. The band structure information for a 2D material is embedded in the calculation of the electron density. DD is especially suitable to compute transport in microscale experimental devices as well as to understand transport in the diffusive regime [188, 189]. A simplified version (while neglecting the diffusive term) is sometimes recalled in the literature invoking linear response theory. A more sophisticated version is the Hydrodynamic model (HD) that also includes heating effects due to high electric fields. The temperature gradient introduces an additional driving force and the current becomes Jn/p = Jdrif+

Jdiff + ρDT∇T, where ρ is the charge carrier concentration and DT is a thermal

diffusivity. HD is a viable alternative to the Monte-Carlo method when dealing with high-field transport and has been used to study the effect of different substrates on graphene at high-field transport [190], providing also a benchmark with experimental results [191].

3.3.6 Analytical Models

Analytical models are especially useful to provide concise but rigorous theoretical support to experiments. Depending on the main mechanism determining the current in the 2D based device, different procedures must be considered. The ’top of the barrier’ [192] and the ’gradual channel approximation’ [193, 194] are mostly consid-ered to model thermionic FETs, in the ballistic and diffusive regimes respectively, but Boltzmann statistics are many times used to simplify the algebra and find closed expressions [67]. In some cases, the Poisson equation is simplified and replaced by a capacitance network [195]. Richardson’s approach, particularized for 2D materials, has been developed distinguishing the effect of the surface and the edges of flake [196, 197]. It has proven to be useful to explain 2D junctions and diodes. When tunneling phenomena are dominant, the Wentzel-Krammer-Brillouin [198] or Bardeen Transfer

(39)

Figure 3.2 | Comparison of the material and device methods from differ-ent according to their accuracy and computation time. The main approxi-mations/limitations and the strengths of each method are also highlighted.

(40)

Hamiltonian [199] models are more appropriate and interesting interlayer tunneling devices have been analyzed using these approximations.

In Fig. 3.2, we show then summarize the main outcomes of the above-mentioned methods, both regarding material and transport models, and specifically how these methods can be classified in terms of accuracy and computing time, highlighting their main strengths and limitations.

3.4 Modeling of 2D-materials-based devices

3.4.1 Graphene

Graphene is the most investigated 2D crystal. Its study has been covered for almost every physical perspective, ranging from excitons and spin transport to antiferromag-netism in nanoribbons or stacked crystals. Several excellent reviews are focusing on the basic science of graphene [200] up to its application in electronics [201]. The well-known zero bandgap of monolayer graphene jeopardizes its use in FETs and more generally in logic devices, but it has been proposed for RF applications or tunnel-ing transistors [198],[202]. When stacked as in bilayer graphene, the band structure becomes sensitive to the vertical electric field, a property that has been assessed in both thermionic [203] and Tunnel FETs [204]. Heterostructures combining graphene with other 2D materials have attracted intense experimental interest, and different multi-scale studies combining ab-initio methods and atomistic transport have been realized [30]. Transport studies based on Monte Carlo [169–171] and Drift-Diffusion [188] with the scattering rates and mobility obtained from ab-initio have also been performed [205].

3.4.2 Group-IV 2D-Xenes: silicene, germanene, stanene

The structures of these materials are similar to graphene but slightly buckled (see Fig. 1.1), due to their larger interatomic distances, which in turn leads to spin-orbit interaction (SOI) [206]. SOI has been investigated from ab-initio and atomistic methods and it has been shown to open small gaps in these 2D crystals [42, 207]. First-principles calculations have shown that the band structure can be tuned with a vertical electric field, although the required electric fields are so large that would break any known field oxide in the FETs.[207, 208].

An option explored through ab-initio simulations consists of functionalizing the materials to open larger gaps [209]. Nonetheless, when the material is cut in armchair nanoribbons, a linear dispersion relationship has been predicted from ab-initio and tight-binding models showing a periodic gap dependence on the nanoribbon width [207, 210]. Zig-zag nanoribbons, on the contrary, they show topologically protected states [211], localized at the edges [207, 212, 213]. This would enable ideal conduction through free-of-scattering edges.

(41)

3.4. Modeling of 2D-materials-based devices 25 On the basis of edge-state transport, a zig-zag stanene nanoribbon transistor based on semi-metallic states has been studied with atomistic models leveraging the possibility of switching the current via scattering modulation [214]. If the nanoribbon is narrow enough, an antiferromagnetic order might prevail with edges of different spin at conduction band minima and valence band maxima, which can be exploited in TFET as demonstrated for stanene employing multi-scale calculations in [146]. Theoretical investigation of FETs based on these materials has also been discussed using ab-initio and atomistic methods [192, 215] as well as Monte Carlo transport simulations [173] and its mobility estimated from MRT [181].

3.4.3 Transition metal dichalcogenides

TMDs electronic properties have been extensively studied via first-principle methods showing a wide range of behaviors: from metallic NbS2 to insulating HfS2 including

better known semiconducting MoS2, WSe2, MoSe2, just to name a few [216–218].

However, the band structure is determined not only by the atomic composition and the metal coordination but also by layer stacking. TMDs have attracted particular attention for electronics due to their semiconducting bandgap that, besides, under-goes an indirect-to-direct transition when the thickness is reduced from bulk to the monolayer. The bandstructure is quite convoluted: eleven orbitals are believed to be relevant in most situations, 6 p-orbitals from the chalcogen atoms and 5 d-orbitals from the metal, which are responsible for spin-orbit coupling[218]. For this reason, single-orbital tight-binding models fail to represent their complexity and models with up to 11 bands are necessary [219], although the simplest alternatives for monolay-ers based on k · p are available [220]. The heavy electron effective mass in TMDs together with their low in-plane dielectric constant initially placed these materials as good candidates for ultra-short channel FETs [143, 221] and also TFETs [155], but nowadays many more of their properties are being explored and more applications in electronics are found. One of them is the possibility of combining them as building blocks to form lateral or vertical van der Waals heterostructures (LH and VH, re-spectively) with tailor-made characteristics. The degree of heterogeneity of materials may vary significantly: from quantum-engineered structures involving layers with no atomic species in common, completely different band structures and incommensurate lattices, to band-engineered structures. Interestingly, in the case of MoS2, one can

take advantage of the change from the metallic 1T - to the semiconducting crystalline 2H-phase to realize a single-material LH [47]. More recently, their application as memristive devices is attracting attention with some theoretical investigation of their use as phase-change materials [222–224]. In addition to well-documented TMDs, a new sub-family has recently emerged combining noble transition metals, Pt, Pd, and Ni, with chalcogenides that are so-called Noble TMDs [225, 226]. They are pre-dicted to have a strong gap dependence on the number of stacked layers, leading in some cases to a change of electronic phase, from semiconductor to metal. The so-called noble-TMDs are promising contenders to build electronic devices as they are

Multi-scale modeling of 2D-material based devices

Università di Pisa

Dottorato di ricerca

in Ingegneria dell’ Informazione

Multi-scale modeling of

2D-material based devices

Doctoral Thesis

Author:

Marta Perucchini

Supervisors:

Prof. Gianluca Fiori

Prof. Giuseppe Iannaccone

Reviewers:

Prof. Francesco Driussi

Prof. Philippe Dollfus

The Coordinator of the PhD Program:

Prof. Fulvio Gini

Pisa, November 2019

Cycle XXXII

Preface

Abstract

Sommario

List of contributions

Comments on my own contribution

Contents

CHAPTER

1

Introduction

CHAPTER

2

2D materials and 3D structures

2.1

Introduction and chapter outline

2.2

2D layered materials

2.2.1

Graphene and other X-enes

2.2.2

Transition-metal Dichalcogenides (TMDs)

2.2.3

Hexagonal Boron Nitride

2.2.4

Other 2D materials and compounds

2.3

Devices and structures made of 2D materials

a)

b)

c)

2.3.1

Lateral FETs

2.3.2

Vertical heterostructures, tunnel FETs

2.3.3

NDR devices

CHAPTER

3

Modeling methods for 2D materials

3.1

Introduction and chapter outline

3.2

Materials modeling

3.2.1

Quantum Monte Carlo (QMC)

3.2.2

Density Functional Theory

3.2.3

GW

3.2.4

Tight-binding

3.2.5

Maximally Localized Wannier Functions (MLWF)

3.2.6

k·p

3.3

Device modeling

3.3.1

Non Equilibrium Green Functions

3.3.2

Conditional wave function (CWF)

3.3.3