Ballistic electron propagation through periodic few-layer graphene nanostructures

Ballistic electron propagation through periodic few-layer graphene nanostructures

Physica E 84 (2016) 60–70 Contents lists available at ScienceDirect Physica E journal homepage: www.elsevier.com/locate/physe Ballistic electron pr...

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Physica E 84 (2016) 60–70

Contents lists available at ScienceDirect

Physica E journal homepage: www.elsevier.com/locate/physe

Ballistic electron propagation through periodic few-layer graphene nanostructures Daniela Dragoman a,c,n, Iulia Mihalache a,b a

University of Bucharest, Faculty of Physics, P.O. Box MG-11, 077125 Bucharest, Romania National Research and Development Institute in Microtechnologies, Str. Erou Iancu Nicolae 126 A, 077190 Bucharest, Romania c Academy of Romanian Scientists, Splaiul Independentei 54, 050094 Bucharest, Romania b

H I G H L I G H T S

   

Propagation in periodic few-layer graphene structures and/or semiconducting stripes. Transmission coefficient calculation using transfer and interface matrices. Different periodic structures modulate differently the electrical current. Method to model ballistic transport in all-graphene devices.

art ic l e i nf o

a b s t r a c t

Article history: Received 17 January 2016 Received in revised form 16 May 2016 Accepted 24 May 2016 Available online 25 May 2016

We have studied electron propagation in periodic structures containing mono- and few-layer graphene regions and/or semiconducting stripes. The calculation of the transmission coefficient in all cases has been performed using transfer matrices inside regions with the same material/potential energy, as well as interface matrices between regions in which the evolution laws of charge carriers differ. Numerical simulations of the transmission coefficient, as well as of the low-temperature conductance, suggest that different periodic structures modulate differently the electrical current. The obtained results can be used to model ballistic transport in all-graphene devices, in particular in few-layer graphene structures. & 2016 Elsevier B.V. All rights reserved.

Keywords: Few-layer graphene Ballistic transport

1. Introduction Graphene, as the one-atom-thick, thinnest yet two-dimensional material, has drawn scientific attention due to its unique mechanical, optical and electrical properties [1], such as very large carrier mobility [2,3], room-temperature micrometer-scale ballistic transport [4,5], or unexpected thermodynamic stability [6]. These properties are a result of the fact that monolayer graphene is a zerogap material with a band structure consisting of Dirac cones, in which the low-energy chiral charge carriers are described by Diraclike massless fermions. As such, graphene differ from common materials in which charge carriers obey the Schrödinger equation. The specific electronic structure of graphene is the source of unusual quantum effects, such as the Klein paradox [7] or the particular quantum Hall effect and Landau level degeneracy [1]. Moreover, few-layer graphene structures (in particular, bilayer and trilayer structures, i.e. stacks of two and, respectively, three n

Corresponding author. E-mail address: [email protected] (D. Dragoman).

http://dx.doi.org/10.1016/j.physe.2016.05.033 1386-9477/& 2016 Elsevier B.V. All rights reserved.

graphene layers bonded by weak van der Waals interactions) exhibit also unique features determined by the stacking arrangement [8]. For instance, bilayer graphene presenting Bernal (AB-) stacking is still a gapless material (in symmetric structures), but its chiral charge carriers obey a parabolic dispersion relation instead of the linear one in monolayer graphene [9]. As a result, the quantum Hall effect, for example, has different manifestations compared to either common materials or monolayer graphene [1,9]. On the other hand, trilayer graphene stacking can occur in two forms: an (ABA) Bernal arrangement with overlapping linear and quadratic dispersion relations or an (ABC) rhombohedral stacking displaying cubic dispersion [10]. Specific quantum effects like quantum Hall conductivity plateaus have been observed experimentally [11] in this material. Throughout this paper we consider only trilayer graphene with rhombohedral stacking. For the mono-, and few-layer graphene cases mentioned above, the different dispersion relations of charge carriers lead to different behaviors of ballistic charge transport. Electrostatic gates in close proximity to graphene or few-layer graphene create potential barriers that modulate the charge carrier concentration and transport through these structures, controlling also the bandgap

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opening or the energy-band overlap in the few-layer graphene case [12]. A bandgap modulation of few-layer graphene is essential in developing devices such as field-effect transistors [13], organic field-effect transistors [14], non-volatile memory devices [15], or nano-electro-mechanical devices [16]. The aim of this paper is to analyze the ballistic transport of lowenergy electrons in the continuum approach through periodic structures containing mono- and few-layer graphene sheets. Several cases are studied, with the intent of emphasizing the differences in electron transmission between similar configurations involving mono- or few-layer graphene. We start by analyzing periodic gated structures of monolayer, bilayer or trilayer graphene, continue with periodic structures consisting of regions with a different number of graphene layers, and study finally periodic arrangements of mono- or few-layer graphene stripes separated by regions in which electrons satisfy the Schrödinger equation. Apart from describing charge carrier transport in potentially allgraphene devices, such as that in [17], the results of this paper emphasize the importance of boundary conditions in charge carrier transport. In the vast majority of charge transport studies in graphene and few-layer graphene structures, the propagation of carriers through regions with different evolution laws is not taken into consideration. However, when this situation is considered, as should be the case in real graphene-based devices, in which Schrödingerlike carriers enter from and are collected by metallic electrodes, for instance, the propagation across interfaces between regions with Schrödinger- and Dirac-like carriers significantly influences the transmission, traversal time and phase of ballistic carriers [18,19]. The present study generalizes previous results on charge carrier propagation through regions with different evolution laws. In particular, it shows that different periodic structures modulate differently the low-temperature current/conductance. As such, the obtained results are of both fundamental and practical interest.

2. Ballistic transport through periodic gated structures in mono-, bi-, or trilayer graphene Throughout this paper we will be referring to a periodic structure as that shown in Fig. 1, in which the electrons enter from and exit into ungated regions, with no applied potential (in which V¼ 0). The period consists of a region of length d on which the potential energy has a value V and an ungated region of length L, in which V¼ 0. In this section we consider propagation of charge carriers in mono-, bi- or tri-layer graphene on which a periodic gate electrode is patterned. In this case the input, output, and the periodic regions refer to the same material and we assume that the potential V can be tuned by biasing the gate electrode. Then, the transmission coefficient through such a structure with period Λ = L + d depends on the equation satisfied by charge carriers. In monolayer graphene, in the region i, in which the

d

L

y t

r

θ

θ



in 1

V=0

t x

V

2

V=0

V



N

V

V=0

Fig. 1. Schematic representation of the periodic structure.

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potential energy takes a constant value V = Vi , the low-energy electrons obey a time-independent Dirac equation

⎛ ⎛ ψ m⎞ 0 kim − ik ym ⎞ ⎛ ψ1mi ⎞ ⎟⎜ ⎟ = (E − Vi ) ⎜ 1i ⎟ ℏvF ⎜⎜ m ⎜ ⎜ ψ m⎟ ⎟ m⎟ m 0 ⎝ 2i ⎠ ⎝ ki + ik y ⎠ ⎝ ψ2i ⎠ for which the solution of ψmT = (ψ1m, ψ2m ) can be written as

the

(1)

spinorial

wavefunction

m m ⎞ ⎛ ⎛ ψ m⎞ Ai exp (ik i x ) + Bi exp ( − ik i x ) ⎟ ⎜ 1i ⎟ = exp (ik ym y ) ⎜ ⎜ ψ m⎟ ⎜ s [A exp (ik m x + iθ m ) − B exp ( − ik m x − iθ m )]⎟ i ⎝ 2i ⎠ i i ⎠ ⎝ i i i i

where

vF is the Fermi velocity,

si = sgn (E − Vi ),

(2) k ym

and

k im = (E − Vi )2 /ℏ2vF2 − (k ym )2 are, respectively, the constant y- and x-component of the electron wavevector, with wavenumber (E − Vi ) /ℏvF , and θim = arctan (k ym/k im ). On the other hand, in a similar ith region in bilayer graphene the spinorial wavefunction ψbT = (ψ1b, ψ2b ) is a solution of the equation

⎛ ⎛ ψ b⎞ 0 (kib − ik yb )2 ⎞ ⎛ ψ1bi ⎞ ℏ2 ⎜ ⎟ ⎜ ⎟ = (E − V ) ⎜ 1i ⎟ i ⎜ ψ b⎟ ⎟⎜ ψ b⎟ 2Mb ⎜⎝ (kib + ik yb )2 0 ⎝ 2i ⎠ ⎠ ⎝ 2i ⎠

(3)

and has the form ⎛ ⎞ ⎛ ψ b⎞ b b Ai exp (ik i x ) + Bi exp ( − ik i x ) ⎟ ⎜ 1i ⎟ = exp (ik b y ) ⎜ y b b ⎜ ⎜ ψb⎟ b b ⎟ ⎝ 2i ⎠ ⎝ si [Ai exp (ik i x + i2θi ) + Bi exp ( − ik i x − i2θi )]⎠

where Mb =

γ /2vF2

is the effective mass and

(4)

γ is the interlayer

tunneling amplitude, k yb and k ib = 2Mb (E − Vi ) /ℏ2 − (k yb )2 are, as before, the constant y- and x-component of the electron wavevector, respectively, with wavenumber 2Mb (E − Vi ) /ℏ2, and b b b θi = arctan (k y /k i ). Finally, for trilayer graphene the charge carriers obey the equation



⎛ ⎛ψt ⎞ (kit − ik ty )3⎞ ⎛ ψ1ti ⎞ 0 ℏ3vF3 ⎜ ⎟ ⎜ ⎟ = (E − Vi ) ⎜ 1i ⎟ ⎜ ⎟ ⎜ψt ⎟ ⎟ ⎜ 2 t γ ⎝ (kit + ik ty )3 0 ⎝ 2i ⎠ ⎠ ⎝ ψ2i ⎠

(5)

and the spinorial wavefunction ψtT = (ψ1t , ψ2t ) can be expressed as t t ⎛ ⎞ ⎛ ψt ⎞ Ai exp (ik i x ) + Bi exp ( − ik i x ) ⎟ ⎜ 1i ⎟ = exp (ik t y ) ⎜ y ⎜ t t ⎜ ψt ⎟ ⎟ t t ⎝ 2i ⎠ ⎝ si [Ai exp (ik i x + i 3θi ) − Bi exp ( − ik i x − i 3θi )]⎠

(6)

with k yt and k it = [(E − Vi ) γ 2]2/3 /ℏ2vF2 − (k yt )2 , the y- and x-components of the electron wavevector with wavenumber [(E − Vi ) γ 2]1/3 /ℏvF , and θit = arctan (k yt/k it ). In order to calculate the transmission coefficients, we use throughout this paper the transfer matrix method. The transfer matrices for mono-, bi- and trilayer graphene between two planes x¼ const. and x + L i ¼ const. in a region i with constant potential energy follow from the form of the corresponding wavefunctions and are given, respectively, by (the subscript 0 refer to wavefunctions at x ¼const.): ⎛ cos (k m L + θ m ) − (i /s ) sin (k m L )⎞ ⎛ ψ m ⎞ ⎛ ψ m⎞ ⎛ ψ m⎞ i i i i i 1 i ⎜ ⎟ ⎜ 1i ⎟ , ⎜ 1i ⎟ = M Lm ⎜ 1i ⎟ = ⎜ ψ m⎟ m m i ⎜ m⎟ ⎜ ⎟ ⎜ m⎟ m ⎝ 2i ⎠0 ⎝ ψ2i ⎠ cos θi ⎝ − (isi ) sin (k i L i ) cos (k i L i − θim ) ⎠ ⎝ ψ2i ⎠

(7)

⎞⎛ b ⎞ ⎛ b b ⎛ ψ b⎞ ⎛ b⎞ b 1 ⎜ sin (k i L i + 2θi ) − (1/si ) sin (k i L i )⎟ ⎜ ψ1i ⎟ ⎜ 1i ⎟ = M b ⎜ ψ1i ⎟ = Li ⎜ ⎟ ⎜ b ⎟, ⎜ b ⎜ ψb⎟ b⎟ b b sin (2θib − k i L i ) ⎠ ⎝ ψ2i ⎠ ⎝ 2i ⎠0 ⎝ ψ2i ⎠ sin (2θi ) ⎝ si sin (k i L i )

(8)

⎛ cos (k t L + 3θ t ) − (i /s ) sin (k t L )⎞ ⎛ t ⎞ ⎛ ψt ⎞ ⎛ ψt ⎞ i i i i i ⎟ ψ1i 1 i ⎜ ⎜ 1i ⎟ = M t ⎜ 1i ⎟ = ⎜ ⎟. Li ⎜ t ⎟ ⎜ ψt ⎟ ⎜ ⎟⎜ t ⎟ t t t cos 3 ( θ ) ψ sin cos 3 is k L k L − ( ) ( ) ( − θit ) ⎠ ⎝ ψ2i ⎠ ⎝ 2i ⎠0 ⎝ 2i ⎠ i ⎝ i i i i i

(9)

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Fig. 2. Dependence of the transmission coefficient of monolayer graphene on energy and (a) gate potential, (b) incidence angle, and (c) width of the gated region.

Once these transfer matrices are known, the total transfer matrix can be calculated by simply multiplying the matrices of all regions. Note that the components of the spinorial wavefunctions are continuous at the interface between regions with different potential energy values in all cases. Then, assuming that electrons are incident from and are collected in ungated regions, in which they propagate at an angle θ, the transmission coefficients in the material k¼ m, b, t (mono-, bior trilayer graphene) can be written, respectively, as

Tm (E, θ ) =

4 (cos θ )2 , m m m m 2 |m11 exp ( − iθ ) + m22 exp (iθ ) + m12 + m21 |

(10)

Tb (E, θ ) =

4 [ sin (2θ )]2 , b b b b 2 | − m11 exp ( − i2θ ) + m22 exp (i2θ ) − m12 + m21 |

(11)

Tt (E, θ ) =

4 [ cos (3θ )]2 , t t t t 2 |m11 exp ( − i3θ ) + m22 exp (i3θ ) + m12 + m21 |

(12)

where

mijk ,

i, j ¼1, 2 are the elements of the total transfer matrices

k in material k. In particular, for periodic structures with N gate Mtot k electrodes, Mtot = (Mdk MLk )N − 1Mdk . Note that the definitions of the transmission coefficients themselves depend on the materials in the input and output regions of the periodic structure. Note also that the transmission coefficients for mono- and bilayer graphene are well known [7], explicit forms of the corresponding transfer matrices appearing in Refs. [20] and [21]; the case of mono- and bilayer graphene superlattices is treated in [22]. The transmission of chiral charge carriers through potential barriers in trilayer graphene has been discussed in [23,24]. The dependence of the transmission coefficient on the energy and gate potential for an incident angle of 25° and d ¼20 nm is represented in Fig. 2(a), the dependence of the same parameter on the incidence angle and energy for V ¼40 meV and d¼ 20 nm being illustrated in Fig. 2(b), while the simulations in Fig. 2(c) were performed for θ ¼25° and V¼40 meV. In all cases it was assumed that the periodic structure with N ¼10 gate electrodes is patterned on monolayer graphene with L ¼20 nm. Corresponding contour plots for the same values of parameters are represented in Figs. 3 (a)–(c) and 4(a)–(c) for bilayer and trilayer periodically gated structures, respectively. Throughout this paper, the simulations

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Fig. 3. Dependence of the transmission coefficient of bilayer graphene on energy and (a) gate potential, (b) incidence angle, and (c) width of the gated region.

were limited to the region in which the low-energy approximation is valid [25]. The results show that, as expected, for V¼ 0 the transmission coefficient in Figs. 2(a), 3(a) and 4(a) is unity, and minibands form as a result of the superlattice effect. These minibands are more apparent for the bi- and trilayer periodic structures. The number and width of the minibands depend on the parameters E, V, θ, and d. The extended regions in which T¼ 0 correspond in all figures to the inhibition of electron propagation, i.e. the occurrence of imaginary values for the x-component of the wavevector. An important conclusion from Figs. 2–4, is that the transmission coefficient has specific behaviors as a function of the number of graphene layers, suggesting that the electrical current could be modulated in different ways by such periodic gated structures.

3. Ballistic transport through periodic structures containing mono-, bi- and trilayer graphene Experiments show that it is nowadays possible to fabricate

structures containing successions of graphene stripes with different number of layers [17]. We examine in this section the ballistic transport through such periodic structures, consisting of monolayer, bilayer and trilayer stripes, with ingoing and outgoing regions of the same material (and the same vanishing potential energy) as the ungated region of width L. The situation treated in this section is more complex since we must take into account the different equations satisfied by charge carriers in adjacent stripes. The consequence is that interface matrices between different regions must be introduced in order to assure the continuity of the probability current density throughout the structure. At the interface between the ith monolayer stripe and the (i þ1)th bilayer stripe, for example, the interface matrix can be found [21] by imposing the requirements that ψ1mi = ψ1,bi + 1, and vF ψ2mi = (ℏk ib+ 1/Mb ) ψ2,b i + 1, and is given by

⎛1 ⎞ 0 ,b ⎟. =⎜ Mim ,i+1 b ℏ ( ) 0 k / M v ⎝ b F ⎠ i+1

(13)

Then, the interface matrix between an ith bilayer stripe and an (iþ1)th monolayer stripe is the inverse of the matrix above. A

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Fig. 4. Dependence of the transmission coefficient of trilayer graphene on energy and (a) gate potential, (b) incidence angle, and (c) width of the gated region.

discussion on the boundary conditions between monolayer and bilayer graphene considering a four-dimensional spinor wavefunction in bilayer can be found in [26–29]. In a similar way, the interface matrix between the ith monolayer stripe and the (i þ1)th trilayer stripe can be determined from the conditions ψ1mi = ψ1,t i + 1, and vF ψ2mi = [ℏ2vF3 (k it+ 1)2 /γ 2] ψ2,t i + 1, and has the form

⎛1 ⎞ 0 ,t ⎟, Mim =⎜ ,i+1 2 t ⎝ 0 ℏ2vF (ki + 1)2 /γ 2 ⎠

(14)

while the interface matrix between bilayer and trilayer stripes, found analogously, by imposing the requirements that ψ1bi = ψ1,t i + 1 and (ℏk ib/Mb ) ψ2bi = [ℏ2vF3 (k it+ 1)2 /γ 2] ψ2,t i + 1, can be expressed as

⎛1 ⎞ 0 ⎟. Mib, i,+t 1 = ⎜ 3 t 2 /(γ 2k b ) 0 M v k ℏ ( ) ⎝ b F i+1 i ⎠

(15)

In both last cases, propagation through a trilayer/monolayer or trilayer/bilayer interface requires multiplication with the inverse of the respective matrices. If the studied periodic structure consists of monolayer graphene with intercalated bilayer stripes, we found that the

transmission coefficient, calculated as in (10) but with a total m transfer matrix Mtot = (M m, bMdb (M m, b)−1MLm )N − 1M m, bMdb (M m, b)−1, vanishes for all values of the parameters E, V, θ and d. This result is quite unexpected since electron propagation through a bilayer graphene region surrounded by monolayer graphene, including the boundary conditions, has been considered before [21] and the transmission coefficient was not found to vanish in general. However, a closer inspection of the matrix of periodic structures shows that, in our case, the trace of the matrix of the period is always complex, so that an effective real propagation constant cannot be derived by using the formalism of Chebyshev polynomials [30]. Therefore, no allowed minibands for electron propagation occur in periodic monolayer/bilayer structures, at least in the ballistic low-energy regime. A similar result has been obtained for the structure consisting of trilayer graphene with intercalated bilayer stripes, these findings illustrating the difference between one stripe of different few-layer graphene sheet and periodic structures. The vanishing transmission coefficient in both cases is due to the different symmetries of the evolution laws for electrons in bilayer and mono- or tri-layer graphene. This symmetry difference leads also to the well-known fact that at normal incidence the transmission coefficient is always unity through a gated

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Fig. 5. Dependence of the transmission coefficient of monolayer graphene intercalated with trilayer stripes on energy and (a) gate potential, (b) incidence angle, and (c) width of the gated region.

monolayer graphene region, irrespective of the barrier height, and always vanishes at propagation through a gated bilayer region with a potential barrier higher than the charge carrier energy [7]. Then, the only structures in which a significant transmission coefficient can be expected are those consisting of monolayer graphene with intercalated trilayer stripes or vive-versa. The first situation is considered in Fig. 5(a)–(c), in which it was assumed that N ¼ 10 and L ¼20 nm. The simulations in Fig. 5 (a) were performed for θ ¼ 10° and d ¼20 nm, those in Fig. 5 (b) for V ¼40 meV and d ¼ 20 nm, while the results in Fig. 5 (c) were obtained for θ ¼ 10° and V ¼40 meV. The periodicity of the structure leads to the appearance of minibands, but the position and width of these minibands change significantly if the monolayers and trilayers switch positions, as can be seen from Fig. 6(a)–(c). These figures are obtained using the same parameters as for Fig. 5(a)–(c), but now the incident, outgoing and ungated regions are trilayer graphene, while monolayer graphene is the layer with width d in Fig. 1. In this case, the transmission coefficient is calculated using (12), with a total t transfer matrix Mtot = ((M m, t )−1Mdm M m, tMLt )N − 1(M m, t )−1Mdm M m, t . As can be seen from Figs. 5 and 6, the transmission coefficient

vanishes over a much wider range of simulation parameters when the incident, outgoing and ungated regions are trilayer graphene, at least in these examples. This result suggests that not only the materials forming the period are important, but also those in the input and output regions, since the transmission coefficient definition depends on them. Note that for ungated monolayer and trilayer graphene V ¼0, but non-vanishing V values can be obtained by gate electrodes; the simulations have considered this more general situation.

4. Ballistic transport through periodic graphene structures separated by Schrödinger-type materials An example of a periodic structure containing graphene and dielectric materials can be found in [31]. It was used for optical and optoelectronic applications, but it could be of interest also for nanoelectronic devices. Therefore, the simulation of such a periodic structure containing regions in which the charge carriers with mass MS satisfy the time-independent Schrödinger equation

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Fig. 6. Dependence of the transmission coefficient of trilayer graphene intercalated with monolayer stripes on energy and (a) gate potential, (b) incidence angle, and (c) width of the gated region.



ℏ2 2 ∇ Ψ = (E − V ) Ψ . 2MS

(16)

is of interest. Since in this case the continuity conditions for charge carriers propagating along the x direction imposes that Ψ and ^ M are constant, the transfer matrix along a distance L in the ∇Ψ ⋅x/ S i ith region in which the potential energy is constant is ⎛ ⎞ ⎛ ⎞ Ψi Ψi ⎜ ⎟ = MS⎜ ⎟ ⎜ ik S Ψ / M ⎟ i ⎜ ik S Ψ / M ⎟ S S ⎝ i i ⎠0 ⎝ i i ⎠ ⎛ ⎞ ⎞ Ψi (M S/ k S ) sin (k S L i ) ⎟ ⎛ cos (k S L i ) ⎜ i i i ⎜ ⎟ =⎜ ⎟⎜ S SL ) ⎜ − (k S/ M ) sin (k S L ) ⎟ ⎝ ik Ψi/ M S ⎟⎠ ( k cos i S ⎝ ⎠ i i i i i

wavefunction, and between the second component of the spinorial wavefunction multiplied with vF , ℏk ib/Mb , and [ℏ2vF3 (k it )2 /γ 2], respectively, for the case of mono-, bi, and tri-layer graphene, and ^ M . Then, the respective interface matrices are: ∇Ψ ⋅x/ S

⎛1 0 ⎞ ,S Mim ⎟, ,i+1 = ⎜ ⎝ 0 ℏ/(ivF )⎠

(18)

⎛1 ⎞ 0 ⎟, Mib, i,+S 1 = ⎜ ⎝ 0 Mb/(ikib )⎠

(19)

⎛1 ⎞ 0 ⎟. Mit,,iS+ 1 = ⎜ t 2 3 2 ⎝ 0 γ (ki ) /(iℏvF )⎠

(20)

(17)

where k iS = 2MS (E − Vi ) /ℏ2 − (k yS )2 and k yS are the x- and y-components of the electron wavevector. In a similar manner as in the previous section, the conservation of probability current density across the interface between the ith few-layer graphene region and the (iþ 1)th region described by the Schrödinger equation can be ensured if the following equalities are imposed at the interface: between the first component of the spinorial wavefunction and the scalar Schrödinger

Note that the matrix in (18) has been already derived in [18], in the context of studying charge carrier transport at the electrode/ graphene and graphene/electrode interfaces. We have performed simulations for the cases of monolayer, bilayer and trilayer graphene interrupted by semiconducting regions of width d in which electrons obey the Schrödinger equation.

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Fig. 7. Dependence of the transmission coefficient of monolayer graphene intercalated with semiconducting stripes on energy and (a) gate potential, (b) incidence angle, and (c) width of the gated region.

In this case V represents the offset of the band conduction in semiconductors with finite bandgap with respect to the Dirac point in few-layer graphene. As in the previous section, after calculating the matrices for a period, we have observed that the matrix for the structure containing bilayer graphene has a complex trace, which implies that formation of allowed minibands for charge carrier propagation is impeded, and so the transmission coefficient practically vanishes. The explanation of this fact relies again on the different symmetries of the electron evolution laws in the low-energy regime in bilayer graphene and semiconductor. As a result, we focused on structures containing monolayer or trilayer graphene stripes separated by semiconducting regions. The corresponding simulation results are represented in Figs. 7(a)– (c) and 8(a)–(c), respectively, for a periodic structure with N ¼10 semiconducting stripes and L ¼20 nm, the effective electron mass in semiconductors being 0.2m0 , with m0 the free electron mass. Figs. 7(a) and 8(a) were obtained for θ ¼ 10°, and d¼ 4 nm, Figs. 7 (b) and 8(b) for d ¼4 nm, V¼40 meV, whereas the parameters used in Figs. 7(c) and 8(c) were θ ¼10°, V ¼40 meV. As expected, in

both cases minibands form due to the periodicity of the structure. However, the shape of the transmission coefficient dependence on V, θ, d and E is determined by the number of graphene layers. In particular, the width of minibands decreases with increasing V and d in both situations, whereas for periodic structures involving trilayer graphene the transmission coefficient vanishes for V ¼E, no such behavior being observed for structures containing monolayer graphene. This result is related to the different propagation laws of ballistic electrons in these cases.

5. Discussions and conclusions The simulations of ballistic charge carrier propagation in the low-energy regime in the previous sections showed that different periodic structures containing mono- and few-layer graphene have different energy dependences of the transmission coefficient. However, in experiments the relevant parameter is not the transmission coefficient but the electrical conductance. According to the Landauer formula, for charge carriers with Fermi energy EF

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Fig. 8. Dependence of the transmission coefficient of trilayer graphene intercalated with semiconducting stripes on energy and (a) gate potential, (b) incidence angle, and (c) width of the gated region.

the linear response conductance per unit width is defined as [26]:

G (EF ) = − G 0

∫ T¯ (E) ∂f (E∂E− EF ) dE

(21)

(e 2 / h )

where G0 = gs gv is the quantum conductance, with gs and gv the spin and valley degeneracies, respectively, f (E ) is the FermiDirac distribution function and

T¯ (E ) = π −1

π /2

∫−π/2 T (E, θ ) dθ

(22)

is the angular averaged transmission probability. To find out how dissimilar the normalized conductances of the different periodic structures studied before are, we represented in Fig. 9 the dependence of this parameter on the Fermi energy. The red, black and blue lines in Fig. 9(a) correspond to periodic gated structures on monolayer, bilayer and trilayer graphene, respectively, with N ¼10, V¼ 40 meV and L ¼d ¼20 nm, the red and blue lines in Fig. 9(b) refer to monolayer graphene intercalated with trilayer stripes and trilayer graphene intercalated with monolayer

stripe, respectively, for the same parameters as above, and the red and blue lines in Fig. 9(c) represent the dependence of the normalized conductance on the Fermi energy for monolayer graphene intercalated with semiconducting stripes and trilayer graphene intercalated with semiconducting stripes, respectively, for N ¼10, V¼40 meV, L ¼20 nm and d ¼4 nm. In all cases the solid lines correspond to conductances at 0 K, while dotted lines are results at 5 K. The simulations in Fig. 9 shows that the conductance/electrical current is modulated differently by different periodic structures, the modulation being in agreement with the dependence of the transmission coefficients on energy and angle calculated in the previous sections. As expected, the oscillatory features in the conductance are smoothed out as the temperature increases due to effect of the Fermi-Dirac distribution function. It should be mentioned that when scattering occurs, i.e. in the non-ballistic region, the differences between the transmission coefficients/ conductances in different periodic structures tend to disappear because of the loss of coherence at propagation. However, in highquality graphene devices, ballistic transport could attain 15 μm below 40 K [32].

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Fig. 9. Dependence of the conductance on the Fermi energy for (a) periodic gated structures consisting of monolayer (red lines), bilayer (black lines) and trilayer graphene (blue lines), (b) monolayer graphene intercalated with trilayer stripes (red lines) and trilayer graphene intercalated with monolayer stripe (blue lines), and (c) monolayer graphene intercalated with semiconducting stripes (red lines) and trilayer graphene intercalated with semiconducting stripes (blue lines). Solid lines are 0 K conductances, while dotted lines are results at 5 K. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

In this paper we have studied the propagation of charge carriers in the low-energy limit in periodic structures containing mono- and few-layer graphene, graphene stripes with different layers and/or semiconducting regions. Note that the low-energy limit model is in agreement with many experiments, including the initial experiments on graphene, which evidenced the linear dispersion relation of this material as well as its unique transport properties [1]. Further experiments show that simulations based on the low-energy ballistic regime are relevant to transport characteristics of graphene-based devices, even at room temperature [33]. Moreover, scanning tunneling spectroscopy experiments reveal that the linear

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dispersion relation in graphene holds up to 200 meV [34], but electron–electron interactions renormalize/enhance the Fermi velocity near the Dirac point; a different value of the Fermi velocity in the transfer matrices can account for this effect. Similar conclusions can be drawn from infrared spectroscopy data [35]. The bandstructure of bilayer graphene is also consistent with the singleparticle tight-binding model even for hot charge carriers [36], whereas absorption in high magnetic fields revealed that deviations from the ideal behavior of Dirac particles in few-layer graphene is observed only for energies higher than 500 meV [37]. So, the lowenergy model can be used to design graphene-based devices, especially because nanoelectronics is concerned with low-power consumption devices. Moreover, the low-energy continuum approximation used throughout this paper has the advantage of unifying the treatment of propagation across interfaces between mono- and few-layer graphene and/or semiconducting regions, through the use of 2  2 interface matrices, at the expense, however, of not being able to capture the influence of the type of edge on the transmission. On the other hand, graphene flakes are not commonly obtained with precise edge types, despite some recent works in this direction [38,39]. Note that the assumption of atomistically sharp interfaces between regions is a common hypothesis in simulations and is also supported by some experiments (see, for example Ref. [40], which reveals sharp changes in conductance between regions with a different number of layers). This assumption is not always accurate, especially at gate-induced potential barrier steps in graphene, where finite p–n junction widths are required to reproduce experimental results on ballistic propagation [41]. However, recent technological advances in fabricating graphene junctions aim to produce high-quality devices with potential barrier interfaces that tend to resemble as close as possible atomic interfaces [42–44]. On the other hand, atomic sharp interfaces occur between graphene and semiconductors such as InSe [45], Si [46] or h-BN [47]. The formalism presented in this paper can accommodate also the cases of non-abrupt interfaces, by dividing the non-rectangular potential barriers into thinner regions in which the potential barrier can be assumed constant; each of this region is then described by a transfer matrix [48]. The design approach of all-graphene devices [17] and periodic graphene-based heterostructures [31] presented in this paper is timely because such devices have already been produced. In addition, high-resolution graphene nanopatterning is within technological reach [49], and controlled etching of exactly one graphene layer per cycle is now possible, such that graphene flakes with an accurate number of layers can be fabricated [50]. Moreover, precisely controlled (by scanning tunneling microscopy) and reversible graphene oxidation/inducing of localized potential barriers can be realized [51]. The transfer matrix method presented in this paper was applied for periodic structures, because such structures have multiple applications (see, for example, [52] and the references therein), but the formalism can be used for any heterostructures involving layers with different charge carrier dispersion relations. Although some studies on periodic structures containing especially mono- and bilayer graphene exist in the literature, this work generalizes previous results since it applies also to periodic gated trilayer graphene, and to structures containing stripes of graphene with different number of layers and/or semiconducting materials. The periodicity of the structure implies that allowed minibands do not form when bilayer graphene and other materials with a different symmetry of the chiral evolution law are involved. In these cases, the trace of the transfer matrix for one period is a complex number. In all other situations, the dependence of the transmission coefficient on the parameters of the periodic structure reveals a strong dependence on the materials that compose the period and those forming the input and output regions,

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since the transmission coefficient definitions depend on the latter. The incorporation of interface matrices in the calculation of the transmission coefficient in structures containing materials obeying different evolution laws is necessary for the conservation of the probability current distribution. The method outlined in this paper for calculating the transmission coefficient and the conductance can be applied for the study of all-graphene devices containing regions with different numbers of graphene layers and/or semiconducting stripes. Although in real devices the simplified assumptions used in the present model may not be valid, we believe that the transport behavior of different periodic structures will still be different and that the transfer matrix formalism is still useful and can at least provide a guideline for the behavior of real structures.

Acknowledgment Iulia Mihalache acknowledges the support of the CNCS-UEFISCDI grant, Project no. PNII-ID-PCCE-2011-2-0069.

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