Single-layer and bilayer graphene superlattices: collimation, additional Dirac points and Dirac lines
SSingle-layer and bilayer graphenesuperlattices: collimation, additionalDirac points and Dirac lines
By Micha¨el Barbier, Panagiotis Vasilopoulos, and Franc¸ois M.Peeters Department of Physics, University of Antwerp,Groenenborgerlaan 171, B-2020 Antwerpen, Belgium Department of Physics, Concordia University,7141 Sherbrooke Ouest, Montr´eal, Quebec, Canada H4B 1R6
We review the energy spectrum and transport properties of several types of one-dimensional superlattices (SLs) on single-layer and bilayer graphene. In single-layergraphene, for certain SL parameters an electron beam incident on a SL is highlycollimated. On the other hand there are extra Dirac points generated for other SLparameters. Using rectangular barriers allows us to find analytic expressions forthe location of new Dirac points in the spectrum and for the renormalization of theelectron velocities. The influence of these extra Dirac points on the conductivityis investigated. In the limit of δ -function barriers, the transmission T through,conductance G of a finite number of barriers as well as the energy spectra of SLsare periodic functions of the dimensionless strength P of the barriers, P δ ( x ) = V ( x ) / (cid:126) v F , with v F the Fermi velocity. For a Kronig-Penney SL with alternating signof the height of the barriers the Dirac point becomes a Dirac line for P = π/ nπ with n an integer. In bilayer graphene, with an appropriate bias applied to thebarriers and wells, we show that several new types of SLs are produced and two ofthem are similar to type I and type II semiconductor SLs. Similar as in single-layergraphene extra “Dirac” points are found. Non-ballistic transport is also considered. Keywords: graphene; electron transport; two-dimensional crystals
1. Introduction
Since the experimental realisation of graphene (Novoselov et al. , 2004) in 2004,this one-atom thick layer of carbon atoms has attracted the attention of the scien-tific world. This attraction pole was created by the prediction that the carriers ingraphene behave as massless relativistic fermions moving in two dimensions. Thelatter particles, which are described by the Dirac-Weyl Hamiltonian, possess inter-esting properties such as a gapless and linear-in-wave vector electronic spectrum,a perfect transmission, at normal incidence, through any potential barrier, i.e., theKlein paradox (Klein, 1929; Katsnelson et al. , 2006; Pereira Jr et al. , 2010; Roslyak et al. , 2010), which was recently addressed experimentally (Young & Kim, 2009;Huard et al. , 2007), the zitterbewegung (Schliemann et al. , 2005; Winkler et al. ,2007; Zawadzki, 2005), etc., see Ref. (Castro Neto et al. , 2009) and (Abergel et al. ,2010) for recent reviews. On the other hand, in bilayer graphene the carriers exhibit
Article submitted to Royal Society
TEX Paper a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n M. Barbier, P. Vasilopoulos, and F. M. Peeters a very different but extraordinary electronic behaviour, such as being chiral (Mc-Cann, 2006; Katsnelson et al. , 2006) but with a different pseudospin (=1) than insingle-layer graphene (=1/2). Although their spectrum is parabolic in wave vectorand also gapless, it is possible to create an energy gap by applying a perpendicularelectric field on a bilayer graphene sample (Castro et al. , 2007). This allows one toelectrostatically create quantum dots in bilayer graphene (Pereira Jr et al. , 2007 b )and enrich its technological capabilities.In previous work we studied the band structure and other properties of single-layer and bilayer graphene (Barbier et al. , 2008, 2009 b ) in the presence of one-dimensional (1D) periodic potential, i.e., a superlattice (SL). SLs are known tobe useful in altering the band structure of materials and thereby broadening theirtechnological applicability.The already peculiar, cone-shaped band structure of single-layer graphene can bedrastically changed in a SL. An interesting feature is that for certain SL parametersthe carriers are restricted to move along one direction, i.e. they are collimated (Park et al. , 2009 a ). Furthermore, it was found that for other parameters of the SL insteadof the single-valley ( the K or K (cid:48) -point) Dirac cone, “extra Dirac points” appearedat the Fermi level in addition to the original one (Ho et al. , 2009). The latter extraDirac points are interesting because of their accompagning zero modes (Brey &Fertig, 2009) and their influence on many physical properties such as the densityof states (Ho et al. , 2009), the conductivity (Barbier et al. , 2010; Wang & Zhu,2010), and the Landau levels upon applying a magnetic field (Park et al. , 2009 b ;Sun et al. , 2010).One can also obtain “extra Dirac points” in bilayer graphene SLs. The possibilityof locally altering the gap (Castro et al. , 2007) of bilayer graphene by applying abias is another way of tuning the band structure. In this review we classify theseSLs in four types. Another interesting result of applying a bias locally is that signflips of the bias introduce bound states along the interfaces (Martin et al. , 2008;Martinez et al. , 2009). These bound states break the time reversal symmetry andare distinct for the two K and K (cid:48) valleys; this opens up perspectives for valley-filterdevices (San-Jose et al. , 2009).In this review we will use the following methods to describe our findings. Forboth single-layer and bilayer graphene we will use the nearest neighbour, tight-binding Hamiltonian in the continuum approximation, and restrict ourselves tothe electronic structure in the neighbourhood of the K point. We then apply thetransfer-matrix method to study the spectrum of and transmission through variouspotential barrier structures, which we approximate by piecewise constant potentials.We consider structures with a finite number of barriers and SLs.We will study ballistic transport in systems with a finite number of barriers usingthe two-probe Landauer conductance while in a SL (infinite number of barriers) wewill evaluate the spectrum and the diffusive conductivity, i.e., we will study non-ballistic transport.The work is organized as follows. In Sec. 2 we investigate various aspects ofballistic transport through a finite number of barriers on single-layer graphene aswell as the spectrum of SLs, with emphasis on collimation and extra Dirac pointsand their influence on non-ballistic transport. In Sec. 3 we carry on the same stud-ies, whenever possible, on bilayer graphene. In addition, we consider various types Article submitted to Royal Society eview: Single-layer and bilayer graphene SLs
2. Single-layer graphene
We describe the electronic structure of an infinitely large, flat graphene flake bythe nearest-neighbour tight-binding model and consider wave vectors close to theK point. The relevant Hamiltonian in the continuum approximation is H = v F σ · ˆp + V + mv F σ z , with ˆp the momentum operator, V the potential, the 2 × σ = ( σ x σ y ), σ z the Pauli-matrices and v F ≈ m/s the Fermi velocity.Explicitly H is given by H = (cid:18) V + mv F − i v F (cid:126) ( ∂ x − i ∂ y ) − i v F (cid:126) ( ∂ x + i ∂ y ) V − mv F (cid:19) . (2.1)The mass term is in principle zero in the nearest-neighbour, tight-binding modelbut due to interaction with a substrate (Giovannetti et al. , 2007) an effective massterm can be induced and results in the opening of an energy gap. Recently therehave been proposals to induce an energy gap in single-layer graphene, and it isappropriate that we consider this mass term where relevant. In the presence ofa 1D rectangular potential V ( x ), such as the one shown in Fig. 1, the equation( H − E ) ψ = 0 admits (right- and left-travelling) plane wave solutions of the form ψ l,r ( x ) e ik y y with ψ r ( x ) = (cid:18) ε + µλ + i k y (cid:19) e i λx , ψ l ( x ) = (cid:18) ε + µ − λ + i k y (cid:19) e − i λx , (2.2)here λ = [( ε − u ( x )) − k y − µ ] / is the x component of the wave vector, ε = EL/ (cid:126) v F , u ( x ) = V ( x ) L/ (cid:126) v F , and µ = mv F L/ (cid:126) . The dimensionless parameters ε , u ( x ) and µ scale with the characteristic length L of the potential barrier structure.For the single or double barrier system this L will be equal to the barrier widthwhile for a SL it will be its period. Neglecting the mass term one rewrites Eq. (2.2)in the simpler form ψ r ( x ) = (cid:18) s e i φ (cid:19) e i λx , ψ l ( x ) = (cid:18) − s e − i φ (cid:19) e − i λx , (2.3)with λ = [( ε − u ( x )) − k y ] / , tan φ = k y /λ , and s = sgn( ε − u ( x )).( a ) A single or double barrier
The model barriers and wells we consider are shown in Fig. 1(a). It is interestingto look at the tunneling through such barriers, which was previously studied by(Katsnelson et al. , 2006) for a single barrier. This was later extended to massiveelectrons with spatially varying mass (Gomes & Peres, 2008).
Transmission.
To find the transmission T through a square-barrier structureone first observes that the wave function in the j th region ψ j ( x ) of the constantpotential V j is given by a superposition of the eigenstates given by Eq. (2.2), ψ j ( x ) = A j ψ rj + B j ψ lj . (2.4) Article submitted to Royal Society
M. Barbier, P. Vasilopoulos, and F. M. Peeters (a) (b)V(x) x WW b w V V b w W b V b Figure 1. (a) A 1D potential barrier of height V b and width W b . (b) A single unit of apotential well next to a potential barrier. The wave function should be continuous at the interfaces. This boundary conditiongives the transfer matrix N j relating the coefficients A j and B j of region j withthose of the region j + 1 in the manner (cid:18) A j B j (cid:19) = N j +1 (cid:18) A j +1 B j +1 (cid:19) . (2.5)By employing the transfer matrix at each potential step we obtain, after n steps,the relation (cid:18) A B (cid:19) = n (cid:89) j =1 N j (cid:18) A n B n (cid:19) . (2.6)In the region to the left of the barrier we assume A = 1 and denote by B = r thereflection amplitude. Likewise, to the right of the n th barrier we have B n = 0 anddenote by A n = t the transmission amplitude.The transmission probability T can be expressed as the ratio of the transmittedcurrent density j x over the incident one, where j x = v F ψ † σ x ψ . This results in T = ( λ (cid:48) /λ ) | t | , with λ (cid:48) /λ the ratio between the wave vector λ (cid:48) to the right and λ to the left of the barrier. If the potential to the right and left of the barrier is thesame we have λ (cid:48) = λ . For a single barrier the transmission amplitude is given by T = | t | = | N | − , with N ij the elements of the transfer matrix N . Explicitly, t can be written as 1 /t = cos( λ b W b ) − iQ sin( λ b W b ) ,Q = ( ε ε b − k y − µ µ b ) /λ λ b ; (2.7)the indices 0 and b refer, respectively, to the region outside and inside the barrierand ε b = ε − u . A contour plot of the transmission is shown in Fig. 2(a). Weclearly see: 1) T = 1 for φ = 0 which is the well-known Klein tunneling, and 2)strong resonances, in particular for E <
0, when λ b W b = nπ , which describe holescattering above a potential well.In the limit of a very thin and high barrier, one can model it by a δ -functionbarrier V ( x ) / (cid:126) v F = P δ ( x ). Using Eq. (2.7) for t gives (Barbier et al. , 2009 a ) T = 1 / [1 + sin P tan φ ] , (2.8)with tan φ = k y /λ the angle of incidence. Notice that this transmission is inde-pendent of the energy and is a periodic function of P . The latter is very differentfrom the non-relativistic case where T is a decreasing function of P. A contour plotof the transmission is shown in Fig. 2(b) and T = 1 for φ ≈ T ( π − P ) = T ( P ). Article submitted to Royal Society eview: Single-layer and bilayer graphene SLs Figure 2. (a) Contour plot of the transmission through a single barrier with µ = 0, W b = L ,and u b = 10. (b) As in (a) for a single δ -function barrier with µ = 0 and u ( x ) = P δ ( x );the transmission is independent of the energy. (c) As in (a) for two barriers with µ = 0, u b = 10, u w = 0, W b = 0 . L , and W w = L . (d) Spectrum of the bound states vs k y fora single ( L = 1, solid red line), two parallel (dashed blue curves), and two anti-parallel(green dash-dotted curves) δ -function barriers ( L is the inter-barrier distance). For two barriers the system becomes a resonant structure, for which it was foundthat the resonances in the transmission depend mostly on the width W w of the wellbetween the barriers (Pereira Jr et al. , 2007 a ). A plot of the transmission is shownin Fig. 2(c). In the limit of two parallel δ -function barriers of equal strength P weobtain the transmission T = (cid:2) φ (cos λ sin 2 P − s sin λ sin P/ cos φ ) (cid:3) − . (2.9)The case of two anti-parallel δ -function barriers of equal strength is also interesting.The relevant transmission is T = (cid:2) cos λ + sin λ (1 − sin φ cos 2 P ) / cos φ (cid:3) − . (2.10) Conductance.
The two-terminal conductance is given by G ( E F ) = G (cid:90) π/ − π/ T ( E F , φ ) cos φ d φ, (2.11)with G = 2 E F L y e / ( v F h ) for single-layer graphene, and L y the width of thesystem. For a single and double barrier, the transmission through which is plotted Article submitted to Royal Society
M. Barbier, P. Vasilopoulos, and F. M. Peeters in Fig. 2(a) and 2(c), the conductance G is shown in Fig. 3(b) and exhibits multipleresonances despite the integration over the angle φ .Taking the limit of a δ -function barrier leads to G periodic in P and given by G/G = 2 (cid:2) − artanh(cos P ) sin P tan P (cid:3) / cos P. (2.12)For one period G is shown in Fig. 3(a). (a) Figure 3. (a) Conductance G vs strength P of a δ -function barrier in single-layer graphene;the conductance is independent of the energy. (b) Conductance G vs energy for the single(solid blue curve) and double (dashed green curve) square barrier of Fig. 2(a) and 2(c). Bound states.
For k y + µ > ε the wave function outside the barrier (well)becomes an exponentially decaying function of x , ψ ( x ) ∝ exp {±| k x | x } with | k x | =[ k y + µ − ε ] / . Localized states form near the barrier boundaries (Pereira Jr et al. ,2006); however, they are propagating freely along the y -direction. The spectrum ofthese bound states can be found by setting the determinant of the transfer matrixequal to zero. For a single potential barrier (well) it is given by the solution of thetranscendental equation | λ | λ b cos( λ b W b ) + ( k y + µ µ b − ε ( ε − u )) sin( λ b W b ) = 0 . (2.13)In Fig. 4(b) these bound states are shown, as a function of k y , by the dashed blue(red) curves.An interesting structure to study is that of a potential barrier next to a well butwith average potential equal to zero, considered by (Arovas et al. , 2010). This isthe unit cell (shown in Fig. 1(b)) of the SL we will use in Sec. 3 where extra Diracpoints will be found. In Fig. 4(a) the Dirac cone outside the barrier is shown as agrey area, inside this region there are no bound states. Superimposed are grey linescorresponding to the edges of the Dirac cones inside the well and barrier that dividethe ( E, k y ) plane into four regions. Region I corresponds to propagating states insideboth the barrier and well while region II (III) corresponds to propagating states onlyinside the well (barrier). In region IV no propagating modes are possible, neitherin the barrier nor in the well. For high thin barriers, region I will become a thinarea adjacent to the upper cone, converging to the dark green line in the limit ofa δ -function barrier. Figure 4(b) shows that the bound states of this structure arecomposed of the ones of a single barrier and those of a single well. Anticrossingstake place where the bands otherwise would cross. The resulting spectrum is clearlya starter of the spectrum of a SL shown in Fig. 4(d). Article submitted to Royal Society eview: Single-layer and bilayer graphene SLs E L / h v F k y L I IIIII IV (a) E L / h v F k y L (b) (d) Figure 4. (a) Four different regions for a single unit of Fig. 1(b) with u b = 24, u w = 16, W b = 0 . W w = 0 .
6. The green line corresponds to region I in the limit of a δ -functionbarrier. (b) Bound states for a single barrier (dashed blue curves) and well (dashed redcurves) and the combined barrier-well unit (black curves). (c) Contour plot of the trans-mission through a unit with µ = 2, u b = − u w = 20 and W b = W w = 0 .
5; the red curvesshow the bound states. (d) Spectrum of a SL whose unit cell is shown in Fig. 1(b), for k x = 0 (blue curves) and k x L = π/ In the limit of δ -function barriers and wells the expressions for the dispersionrelation are strongly simplified by setting µ = 0 in all regions. For a single δ -functionbarrier the bound state is given by ε = sgn(sin P ) | k y | cos P, (2.14)which is a straight line with a reduced group velocity v y ; the result is shown inFig. 2(d) by the red curve. Comparing with the single-barrier case we notice thatdue to the periodicity in P , the δ -function barrier can act as a barrier or as a welldepending on the value of P .For two δ -function barriers there are two important cases: the parallel and theanti-parallel case. For parallel barriers one finds an implicit equation for the energy | λ (cid:48) cos P + ε sin P | = | e − λ (cid:48) k y sin P | , (2.15)where λ (cid:48) = | λ | , while for anti-parallel barriers one obtains k y sin P = λ (cid:48) / (1 − e − λ (cid:48) ) . (2.16) Article submitted to Royal Society
M. Barbier, P. Vasilopoulos, and F. M. Peeters
For two (anti-)parallel δ -function barriers we have, for each fixed k y and P , twoenergy values ± ε , and therefore two bound states. In both cases, for P = nπ thespectrum is simplified to the one in the absence of any potential ε = ±| k y | . InFig. 2(d) the bound states for double (anti-)parallel δ -function barriers are shown,as a function of k y L , by the dashed blue (dash-dotted green) curves. For anti-parallel barriers we see that there is a symmetry around E = 0, which is absentwhen the barriers are parallel. ( b ) Superlattice
Now we will consider the system of a superlattice with a corresponding 1Dperiodic potential, with square barriers, given by V ( x ) = V ∞ (cid:88) j = −∞ [Θ( x − jL ) − Θ( x − jL − W b )] . (2.17)with Θ( x ) the step function. The corresponding wave function is a Bloch functionand satisfies the periodicity condition ψ ( L ) = ψ (0) exp( ik x ), with k x now the Blochphase. Using this relation together with the transfer matrix for a single unit ψ ( L ) = M ψ (0) leads to the conditiondet[ M − exp( ik x )] = 0 . (2.18)This gives the transcendental equationcos k x = cos λ w W w cos λ b W b − Q sin λ w W w sin λ b W b , (2.19)from which we obtain the energy spectrum of the system. In Eq. (2.19) we used thefollowing notation: ε w = ε + uW b , ε b = ε − uW w , u = V L/ (cid:126) v F , W b,w → W b,w /L,λ w = [ ε w − k y − µ w ] / , λ b = [ ε b − k y − µ b ] / , Q = ( ε w ε b − k y − µ b µ w ) /λ w λ b . Numerical results for the dispersion relation E ( k y ) are shown in Fig. 4(d). Wesee the appearance of bands (green areas) which for large k y values collapse intothe bound states (where the red and blue curves meet) while the charge carriersmove freely along the y direction.( c ) Collimation and extra Dirac points
As shown by various studies, carriers in graphene SLs exhibit several interestingpecularities that result from the particular electronic SL band structure. In a 1D SLit was found that the spectrum can be altered anisotropically (Park et al. , 2008 a ;Bliokh et al. , 2009). Moreover, this anisotropy can be made very large such thatfor a broad region in k space the spectrum is dispersionless in one direction, andthus electrons are collimated along the other direction (Park et al. , 2009 a ). Evenmore intriguing was the ability to split off ”extra Dirac points” (Ho et al. , 2009)with accompanying zero modes (Brey & Fertig, 2009) which move away from theK point along the k y direction with increasing potential strength. Here we willdescribe these phenomena for a SL of square potential barriers. Article submitted to Royal Society eview: Single-layer and bilayer graphene SLs et al. , 2009 a ); sub-sequently we will find the conditions on the parameters of the SL for which acollimation appears. It turns out that they are the same as those needed to createa pair of extra Dirac points. Figure 5. The lowest conduction band of the spectrum of graphene near the K point inthe absence of SL potential (a), (b) and in its presence (c), (d) with u = 4 π . (a) and (c)are contour plots of the conduction band with a contour step of 0 . (cid:126) v F /L . (b) and (d)show slices along constant k y L = 0 , . , . π . Following (Park et al. , 2009 a ) the condition for collimation to occur is (cid:82) BZ e i s ˆ sα ( x ) =0, where the function α ( x ) = 2 (cid:82) x u ( x (cid:48) )d x (cid:48) embodies the influence of the potential, s = sign( ε ) and ˆ s = sign( k x ). For a symmetric rectangular lattice this correspondsto u/ nπ . The spectrum for the lowest energy bands is then given by (Park et al. , 2008 b ) ε ≈ ± (cid:2) k x + | f l | k y (cid:3) / + πl/L (2.20)with f l being the coefficients of the Fourier expansion e i α ( x ) = (cid:80) ∞ l = −∞ f l e i2 πlx/L .The coefficients f l depend on the potential profile V ( x ), with | f l | <
1. For a sym-metric SL of square barriers we have f l = u sin( lπ/ − u/ / ( l u − u / | f l | < y direction v y < v F which can beseen from Eq. (2.20).In Fig. 5(b),(d) we show the dispersion relation E vs k x for u = 4 π at constant k y . As can be seen, when a SL is present in most of the Brillouin zone the spectrum,partially shown in (c), is nearly independent of k y . That is, we have collimation Article submitted to Royal Society M. Barbier, P. Vasilopoulos, and F. M. Peeters of an electron beam along the SL axis. The condition u = V L/ (cid:126) v F = 4 nπ showsthat altering the period of the SL or the potential height of the barriers is sufficientto produce collimation. This makes a SL a versatile tool for tuning the spectrum.Comparing with Figs. 5(a), (b) we see that the cone-shaped spectrum for u = 0, istransformed into a wedge-shaped spectrum (Park et al. , 2009 a ).We will compare this result now with an other approximate result for the spec-trum, where we suppose ε small instead of k y small. We start with the transcenden-tal equation (2.19). As we are interested in an analytical approximate expressionfor the spectrum, we choose to expand the dispersion relation around ε = 0 up tosecond order in ε . The resulting spectrum is ε ± = ± (cid:34) | a | (cid:2) k y sin ( a/
2) + a sin ( k x / (cid:3) k y a sin a + a u / − k y u sin ( a/ (cid:35) / , (2.21)with a = [ u / − k y ] / . In order to compare this spectrum with that by (Park et al. , 2009 a ), we expand Eq. (2.19) for small k and ε ; this leads to ε ≈ ± (cid:2) k x + k y sin ( u/ / ( u/ (cid:3) / . (2.22)This spectrum has the form of an anisotropic cone and corresponds to that ofEq. (2.20) for l = 0 (higher l correspond to higher energy bands). In Fig. 6(a),(b) we see that the cone-shaped spectrum in (a), for u = 0, is transformed intoa anisotropic spectrum in (b), for u = 4 . π , having peculiar extra Dirac points.These extra Dirac points cannot be described by a spectrum having an anisotropiccone-shape, therefore we compare the two approximate spectra. In Fig. 6(c), (d) weshow how Eq. (2.21)) and Eq. (2.22) differ from the “exact” numerically obtainedspectrum. From this figure one can see that Eq. (2.21) describes the lowest bandsrather well for ε <
1, while Eq. (2.22) is sufficient to describe the spectrum nearthe Dirac point. The former equation will be usefull when describing the spectrumnear the extra Dirac points and we will use it to obtain the velocity.We now move on to another important feature of the spectrum, the extra Diracpoints first obtained by (Ho et al. , 2009) using tight-binding calculations. Theseextra Dirac points are found as the zero-energy solutions of the dispersion relationin Eq. (2.19) for zero energy (Barbier et al. , 2010).In order to find the location of the Dirac points we assume k x = 0, ε = 0, µ b = µ w = 0, and consider the special case of W b = W w = 1 / λ/ (cid:2) ( u / k y ) / ( u / − k y ) (cid:3) sin λ/ , (2.23)has solutions for u / − k y = u / k y or sin λ/ k y = 0 (at the Dirac points) as k y,j ± = ± (cid:114) u − j π ; (2.24)the extra Dirac points are for j (cid:54) = 0. For a SL spectrum symmetric around zeroenergy, the extra Dirac points are at ε = 0. We expect from the considerations ofSec. 2(b) (and Fig. 4(b)) that for unequal barrier and well widths this will no longer Article submitted to Royal Society eview: Single-layer and bilayer graphene SLs k x = 0, by (Barbier et al. ,2010) ε j,m = u − W b ) + π u (cid:18) j W w − ( j + 2 m ) W b (cid:19) ,k yj,m = ± (cid:104) ( ε j,m + uW b ) − ( jπ/W w ) (cid:105) / , (2.25)where j and m are integers, and m (cid:54) = 0 corresponds to higher and lower crossingpoints. Also, perturbing the potential with an asymmetric term, as done by (Park et al. , 2009 b ), leads to qualitatively similar results. -6 -4 -2 0 2 4 60123 E L / h v F k y L / π (c) -6 -4 -2 0 2 4 60123 E L / h v F k y L / π (d) Figure 6. The spectrum of graphene near the K point in the absence of a SL (a) and inits presence (b) with u = 4 . π . (c) and (d) The SL spectrum with u = 10 π , the lowestconduction bands are coloured in cyan, red, and green for, respectively, the exact, and theapproximations given by (c) Eq. (2.21) and (d) Eq. (2.22), respectively. The approximatespectra are delimited by the dashed curves. An investigation of the group velocity near the (extra) Dirac points is appropri-ate for understanding the transport of carriers in the energy bands close to zero en-ergy. Near the extra Dirac points the group velocity tends to renormalise differentlyas compared to the original Dirac point. Near them v is oriented along the y direc-tion, while near the latter one v is oriented along the x direction (Ho et al. , 2009).The group velocity near the extra Dirac points can be calculated from Eq. (2.21). Atthe j th extra Dirac point the magnitude of the velocity v /v F = ( ∂ε/∂k x , ∂ε/∂k y )is given by v x /v F = 16 π j cos( k x / /u v y /v F = ( u / − j π ) /u , (2.26) Article submitted to Royal Society M. Barbier, P. Vasilopoulos, and F. M. Peeters while at the main Dirac point it is given by v x /v F = 1 and v y /v F = 4 sin( u/ /u .The dependence of the velocity components on the strength of the potential barriersis shown in Fig. 7. From this figure we observe that new extra Dirac points emergeupon increasing u = V L/ (cid:126) v F (consistent with Eq. (2.24)) and v x decreases while v y increases. The Dirac point itself, however, shows a different behaviour uponincreasing u , namely v x = v F constant, and v y is here a globally decaying functionshowing v y = 0 for periodic values of u , u = 4 nπ , with n a nonzero positive integer. Figure 7. The group velocity components v y and v x at the Dirac point j = 0 (shown,respectively, by the solid blue and the dot-dot-dashed red curve), and at the extra Diracpoints j = 1 , , u = V L/ (cid:126) v F . Conductivity.
We now turn to the transport properties of a SL and look atthe influence of these extra Dirac points on the conductivity. The diffusive dc con-ductivity σ µν for the SL system can be readily calculated from the spectrum if weassume a nearly constant relaxation time τ ( E F ) ≡ τ F . It is given by (Charbonneau et al. , 1982) σ µν ( E F ) = e βτ F A (cid:88) n, k v nµ v nν f n k (1 − f n k ) , (2.27)with A the area of the system, n the energy band index, µ, ν = x, y, and f n k =1 / [exp( β ( E F − E n k )) + 1] the equilibrium Fermi-Dirac distribution function; β =1 /k B T and the temperature enters the results through the dimensionless value for β which is β = (cid:126) v F /k B T L = 20.For comparison we first look at the conductivity tensor at zero temperature andin the absence of a SL. For single-layer graphene the conductivity is given by σ µµ ( ε F ) /σ = ε F / π (2.28)with σ = e / (cid:126) ,In Figs. 8(a), (b) the conductivities σ xx and σ yy are shown for a SL as functionsof the energy. Notice that for small energies the slope of the conductivity σ yy istunable to a large extent by altering the parameter u of the SL. The dashed bluecurves correspond to u = 4 π and the rather flat dispersion in the y direction for Article submitted to Royal Society eview: Single-layer and bilayer graphene SLs σ yy (for energies EL/ (cid:126) v F <
1) compared to the conductivity in the absence of a SL. The solidred curves on the other hand correspond to u = 6 π and due to the extra Diracpoints, which have a rather flat dispersion in the x direction (Ho et al. , 2009), theconductivity σ yy is large. Figure 8. (Color online) Conductivities, σ xx in (a) and σ yy in (b), vs Fermi energy for aSL on single-layer graphene with u = 4 π and 6 π for, respectively, the blue dashed andred solid curves. In both cases W b = W w = 0 .
5. The dash-dotted black curves show theconductivities in the absence of the SL potential, σ xx = σ yy = ε F σ / π . ( d ) Dirac lines
P -P(a) (b)PV(x) x
Figure 9. (a) Schematics of Kronig-Penney SL on single-layer graphene. (b) ExtendedKronig-Penney SL.
In an effort to simplify the expressions for the dispersion relation we replace, aswe did for the few-barrier structures, the SL barriers by δ -function barriers. Thesquare SL potential is then approximated by V ( x ) = P ∞ (cid:88) j = −∞ δ ( x − jL ) . (2.29)This potential leads to the dispersion relationcos k x = cos λ cos P + ( ε/λ ) sin λ sin P, (2.30)which is periodic in P . This is in sharp contrast with that for standard electronswhich is not periodic in P and which in our notation readscos k x = cos λ (cid:48) + ( µP/λ (cid:48) ) sin λ (cid:48) , (2.31) Article submitted to Royal Society M. Barbier, P. Vasilopoulos, and F. M. Peeters where µ = mv F L/ (cid:126) and λ (cid:48) = [2 µε − k y ] / . As can be seen from Fig. 10(a), theenergy band near the Dirac point has the interesting property that it becomes nearlyflat in k x , forming a plane, for large k y . The angle which the asymptotic plane makeswith the zero-energy plane depends on P and the group velocity v y correspondingto this asymptotic plane varies from − v F to v F in each period nπ < P < ( n + 1) π .Notice that no extra Dirac points are found and the reason is the same as thatfor the asymmetric SL potential, i.e., the extra Dirac points shift away from zeroenergy. Alternatively, we can try to shed some light by comparing with Sec. 2(b),where it is explained that the bound states for a single unit of the SL potential aresimilar to those of the combined single barrier and well. In the region where thebound states cross (denoted by I in Fig. 4(a)) anti-crossings occur and correspondingcrossings in the SL spectrum (extra Dirac points) are expected. In the limit of a δ -function barrier this region is reduced to a line (the dark green line in Fig. 4(a)).This prevents anti-crossings from occurring and in this way no extra Dirac pointsare expected. (a) -1 0 1-101 -1 0 1 E L / π h v F k x L / π k y L / π (b) Figure 10. (a) Spectrum for a Kronig-Penney SL with P = 0 . π . The blue and red curvesshow, respectively, the k x = 0 and k x = π/L results which delimit the energy bands (greencoloured regions). (b) Spectrum for an extended Kronig-Penney SL with P = π/
2. Noticethat the Dirac point has become a Dirac line.
Extended Kronig-Penney model.
To re-establish the symmetry between electronsand holes, as in the case of square barriers with W b = W w , we can use alternating-in-sign δ -function barriers. The unit cell of the periodic potential contains one suchbarrier up, at x = 0, followed by a barrier down, at x = L/
2, see Fig. 9(b). Thepotential is given by V ( x ) = P ∞ (cid:88) j = −∞ [ δ ( x − jL ) − δ ( x − jL − L/ , (2.32)and is the asymptotic limit of the potential shown in Fig. 1(b). The resulting transfermatrix leads to the dispersion relationcos k x = cos λ − (2 k y /λ ) sin ( λ/
2) sin P. (2.33)This dispersion relation is periodic in P . As shown in Fig. 10(b) no extra Diracpoints occur, but for the particular case of P = ( n + 1 / π , n an integer, the Article submitted to Royal Society eview: Single-layer and bilayer graphene SLs k y we see that Eq. (2.33) has a solutionwith ε = k x = 0, which means the Dirac point at k x = k y = 0 turned into a Diracline along the k y axis. If we take k y not too large (of the order of k x ), this spectrumhas a wedge structure as was also found for rectangular SLs. For k y → ∞ , though,the spectrum becomes a horizontal plane situated at ε = 0. We can generalize thismodel by taking the distance W between the two barriers of the unit cell not equalto L/
2. This was done by (Ramizani Masir et al. , 2010, unpublished work). Theyfound an approximate analytic expression for the dispersion given by ε ≈ [ k x + F k y ] / , F = W + ( L − W ) + 2 W ( L − W ) cos(2 P ) . (2.34)This dispersion has the shape of an anisotropic cone with a renormalized velocityin the y direction. Comparing with Eqs. (2.20) and (2.22), we observe that thecondition for collimation and the velocity renormalization in the y direction isquite different for square barriers. For instance, in the extended KP model, with W = L/
2, we find v y /v F = | cos P | while for square barriers the result is v y /v F =sin( u/ / ( u/ P ≡ u/
4, the velocity in the y direction is maximum v y = v F for P = (1 / n ) π in the extended KP model whilefor square barriers v y = 0 at these points.
3. Bilayer graphene
We now turn to bilayer graphene and use again the nearest-neighbour, tight-bindingHamiltonian in the continuum approximation with k close to the K point. If weinclude a potential difference between the two layers, the Hamiltonian is given by H = U v F π t ⊥ v F π † U t ⊥ U v F π † v F π U . (3.1)Here U and U are the potentials on layers 1 and 2, respectively, 2∆ = U − U is the potential difference, and t ⊥ describes the coupling between the layers. Theenergy spectrum for free electrons is given by (McCann, 2006; Barbier et al. , 2009 b ) ε = u ± (cid:104) ∆ + k + t ⊥ k + k t ⊥ + t ⊥ / (cid:105) / ,ε = u ± (cid:104) ∆ + k + t ⊥ − (4∆ k + k t ⊥ + t ⊥ / (cid:105) / , (3.2)with u = u + ∆ and u = u − ∆. Contrary to Sec. 3 we use units in inversedistance, namely, ε = E/ (cid:126) v F , u j = U j / (cid:126) v F , and k = [ λ + k y ] / . This spectrumexhibits an energy gap that for 2∆ (cid:28) t ⊥ equals the difference 2∆ between theconduction and valence band at the K point (McCann, 2006).Solutions for this Hamiltonian are four-vectors ψ and for 1D potentials we canwrite ψ ( x, y ) = ψ ( x ) exp( ik y y ). If the potentials U and U do not vary in space,these solutions are of the formΨ ± ( x ) = f ± h ± g ± h ± e ± i λx +i k y y , (3.3) Article submitted to Royal Society M. Barbier, P. Vasilopoulos, and F. M. Peeters with f ± = [ − i k y ± λ ] / [ ε (cid:48) − δ ], h ± = [( ε (cid:48) − δ ) − k y − λ ] / [ t ⊥ ( ε (cid:48) − δ )], and g ± =[ ik y ± λ ] / [ ε (cid:48) + δ ]; the wave vector λ is given by λ ± = (cid:20) ε (cid:48) + δ − k y ± (cid:113) ε (cid:48) δ + t ⊥ ( ε (cid:48) − δ ) (cid:21) / . (3.4)We will write λ + = α and λ − = β .( a ) Tuning of the band offsets
It was shown before that using a 1D biasing, indicated in Figs. 11(a,b,c) by2∆, one can create three types of heterostructures in graphene (Dragoman et al. ,2010). A fourth type, where the energy gap is spatially kept constant but the biasperiodically changes sign along the interfaces, can be introduced (see Fig. 11(d)).We characterize these heterostructures as follows:1)
Type I:
The gate bias applied in the barrier regions is larger than in the wellregions.2)
Type II:
The gaps, not necessarily equal, are shifted in energy but they havean overlap as shown.3)
Type III:
The gaps, not necessarily equal, are shifted in energy and have nooverlap.4)
Type IV:
The bias changes sign between successive barriers and wells but itsmagnitude remains constant.Type IV structures have been shown to localize the wave function at the inter-faces (Martin et al. , 2008; Martinez et al. , 2009). To understand the influence ofsuch interfaces in this section we will separately investigate structures with such asingle interface embedded by an anti-symmetric potential.
Ec,b Ec,wEv,wEv,b
2D 2D w b
Type I
Ec,b Ec,wEv,wEv,b
2D 2D w b
Type II
Ec,b Ec,wEv,wEv,b
2D 2D w b
Type III
2D 2D w b
Type IV
EcEv
Figure 11. Four different types of band alignments in bilayer graphene. E c,b , E c,w , E v,c ,and E v,b denote the energies of the conduction (c) and valence (v) bands in the barrier(b) and well (w) regions. The corresponding gap is, respectively, 2∆ b and 2∆ w . Article submitted to Royal Society eview: Single-layer and bilayer graphene SLs | E | < t ⊥ , the eigenstates whichare propagating are the ones with λ = α . Accordingly, from now on we will assumethat β is complex. In this way we can simply use the transfer-matrix approach ofSec. 2 in the transmission calculations. This leads to the relation t e d = N r e g . (3.5)Again the transmission is given by T = | t | .For a single barrier the transmission in bilayer graphene is given by a com-plicated expression. Therefore, we will first look at a few limiting cases. First weassume a zero bias ∆ = 0 that corresponds to a particular case of type III het-erostructures. In this case we slightly change the definition of the wave vectors: for∆ = 0 we assume α ( β ) = [ ε + ( − ) εt ⊥ − k y ] / . If we restrict the motion along the x axis, by taking k y = 0, and assume a bias ∆ = 0, then the transmission T = | t | is given via 1 /t = e iα D [cos( α b D ) − iQ sin( α b D )] ,Q = 12 (cid:18) α b ε α ε b + α ε b α b ε (cid:19) . (3.6)This expression depends only on the propagating wave vector α ( β for E < < ε < u . Due to the coupling for nonzero k y withthe localized states, resonances in the transmission will occur (see Fig. 12). We caneasily generalize this expression to account for the double barrier case under thesame assumptions. With an inter-barrier distance W w one obtains the transmission(Barbier et al. , 2009 b ) T d = | t d | from t d = e i α ( W w +2 W b ) | t | e i φ t − | r | e i φ r e i α W w , (3.7)with r = | r | e i φ r , and t = | t | e i φ t , corresponding to the single barrier transmissionand reflection amplitudes. In this case we do have resonances due to the well states;they occur for e i2 φ r e i2 α W w = 1. As φ r is independent of W w , one obtains moreresonances by increasing W w .For a single δ -function barrier with potential V ( x ) / (cid:126) v F = P δ ( x ) under zerobias, we find the transmission amplitude1 /t = cos P + i µ sin P + ( α − β ) k y αβε sin P cot P + i ν , (3.8)where µ = ( ε + 1 / /α and ν = ( ε − / /β . Notice that this formula is periodic inthe strength of the barrier P as in the single-layer case.For the general case we obtained numerical results for the transmission throughvarious types of single and double barrier structures; they are shown in Fig. 13. Article submitted to Royal Society M. Barbier, P. Vasilopoulos, and F. M. Peeters (b)
Figure 12. (a) Contour plot of the transmission for the potential of Fig. 1(b) in bilayergraphene with W b = W w = 40 nm, V b = − V w = 100 meV and zero bias. Bound states areshown by the red curves. (b) Spectrum for a SL whose unit is the potential structure ofFig. 1(b). Blue and red curves show, respectively, the k x = 0 and k x = π/L results whichdelimit the energy bands (green coloured regions). The different types of structures clearly lead to different behaviour of the tunnellingresonances.An interesting structure to study is the fourth type of SLs shown in Fig. 11(d).To investigate the influence of the localized states (Martin et al. , 2008; Martinez et al. , 2009) on the transport properties we embed the anti-symmetric potentialprofile in a structure with unbiased layers.
Conductance
At zero temperature G can be calculated from the transmissionusing Eq. (13) with G = (4 e L y / πh ) ( E F + t ⊥ E F ) / / (cid:126) v F for bilayer grapheneand L y the width of the sample. The angle of incidence φ is given by tan φ = k y /α with α the wave vector outside the barrier. Figure 14 shows G for the four SL types.Notice the clear differences in 1) the onset of the conductance and 2) the numberand amplitude of the oscillations. Bound states.
To describe bound states we assume that there are no propagat-ing states, i.e., α and β are imaginary or complex (the latter case can be solvedseparately), and only the eigenstates with exponentially decaying behaviour arenonzero leading to the relation f d e d = N f g e g . (3.9)From this relation we can find the dispersion relation for the bound states.To study the localized states for the anti-symmetric potential profile (Martin et al. , 2008; Martinez et al. , 2009) we will use a sharp kink profile (step function).The spectrum found by the method above is shown in Fig. 15(a). We see that thereare two bound states, both with negative group velocity v y ∝ ∂ε/∂k y , as foundpreviously by (Martin et al. , 2008). No bound state near zero energy was found for k y → ∞ in contradiction with (Martinez et al. , 2009). For zero energy we find the Article submitted to Royal Society eview: Single-layer and bilayer graphene SLs Figure 13. (Color online) Contour plot of the transmission through a single barrier in (a)and (b), for width W b = 50 nm, and through double barriers in (c), (d), (e), and (f) ofequal widths W b = 20 nm that are separated by W w = 20 nm. Other parameters are asfollows: (a) ∆ b = 100 meV, V b = 0 meV. (b) ∆ b = 20 meV, V b = 50 meV. (c) Type I : V b = V w = 0 meV, ∆ w = 20 meV, and ∆ b = 100 meV. (d) Type II : V b = − V w = 20meV, ∆ w = ∆ = 50 meV, (e) Type III : V b = − V w = 50 meV, ∆ w = ∆ b = 20 meV. (f) Type IV : V b = V w = 0 meV, ∆ b = − ∆ w = 100 meV. solution k y = ±
12 [∆ + (∆ + 2∆ t ⊥ ) / ] / ≈ ± (cid:112) ∆ t ⊥ / / , ∆ (cid:28) t ⊥ ; (3.10)the approximation on the second line leads to the expression found by (Martin et al. , 2008). ( b ) Superlattices
The heterostructures above (see Fig. 11), can be used to create four differenttypes of SLs (Dragoman et al. , 2010). We will especially focus on type IV and typeIII SLs in certain limiting cases.
Article submitted to Royal Society M. Barbier, P. Vasilopoulos, and F. M. Peeters
Figure 14. (Color online) Two-terminal conductance of four equally spaced barriers vsenergy for W b = W w = 10 nm and different SL types I-IV. The solid red curve (type I) isfor ∆ b = 50 meV, ∆ w = 20 meV, and V w = V b = 0. The blue dashed curve (type II) isfor ∆ b = ∆ w = 50 meV and V b = − V w = 20 meV. The green dotted curve (type III) isfor ∆ b = ∆ w = 20 meV and V b = − V w = 50 meV. The black dash-dotted curve (type IV)is for ∆ b = − ∆ w = 50 meV and V w = V b = 0. (a) -1.0 -0.5 0.0 0.5 1.0-0.50.00.5 k y hv F / t t E / t t (b) Figure 15. (a) Bound states of the anti-symmetric potential profile (type IV) with bias∆ w = − ∆ b = 200 meV. (b) Contour plot of the transmission through a 20 nm wide barrierconsisting of two regions with opposite biases ∆ = ±
100 meV.
For a type I SL we see in Fig. 16(a) that the conduction and valence band of thebilayer structure are qualitatively similar to those in the presence of a uniform bias.Type II structures maintain this gap, see Fig. 16(b), as there is a range in energyfor which there is a gap in the SL potential in the barrier and well regions. In typeIII structures we have two interesting features, which can close the gap. First wesee from Fig. 12(b) that for zero bias, similar to single-layer graphene, extra Diracpoints appear for k x = 0, likewise for Fig. 4(d). In the case W b = W w = L/ W , k x = 0 and E = 0 the values for the k y where extra Dirac points occur are givenby the following transcendental equation[cos( αW ) cos( βW ) −
1] + α + β − ky αβ sin( αW ) sin( βW ) = 0 . (3.11)Comparing the figures 12(b) and 4(d) we remark that, different from the single-layer case, for bilayer graphene the bands in the barrier region are not only flatin the x direction for large k y values but also for small k y . The latter correspondsto the zero transmission value inside the barrier region for tunneling through a Article submitted to Royal Society eview: Single-layer and bilayer graphene SLs Figure 16. (Color online) Lowest conduction and highest valence band of the spectrumfor a square SL with period L = 20 nm and W b = W w = 10 nm. (a) Type I : ∆ b = 100meV and ∆ w = 0. (b) Type II : As in (a) for ∆ b = ∆ w = 50 meV, and V b = − V w = 25meV. (c) Type III : V b = − V w = 25 meV, and ∆ b = ∆ w = 0. (d) Type III : V b = − V w = 50meV and ∆ b = ∆ w = 0. (e) Type IV : Plot of the spectrum for a square SL with averagepotential V b = V w = 0 and ∆ b = − ∆ w = 100 meV. The contours are for the conductionband and show that the dispersion is almost flat in the x direction. single unbiased barrier in bilayer graphene. Secondly, if there are no extra Diracpoints (small parameter uL ) for certain SL parameters, the gap closes at two pointsat the Fermi-level for k y = 0. The latter we will investigate a bit more in theextended Kronig-Penney model. Periodically changing the sign of the bias (type IV)introduces a splitting of the charge neutrality point along the k y axis; this agreeswith what was found by (Martin et al. , 2008). We illustrate that in Fig. 13(e) for aSL with ∆ b = − ∆ w = 100 meV. We also see that the two valleys in the spectrum arerather flat in the x direction. Upon increasing the parameter ∆ L , the two touchingpoints shift to larger ± k y and the valleys become flatter in the x direction. For allfour types of SLs the spectrum is anisotropic and results in very different velocitiesalong the x and y directions. Extended Kronig-Penney model.
To understand which SL parameters lead tothe creation of a gap we look at the Kronig-Penney limit of type III SLs for zerobias (Barbier et al. k y = 0. This assumption is certainly not valid if the parameter uL is large because in that case we expect extra Dirac points (not in the KP limit)to appear that will close the gap. The spectrum for k y = 0 is determined by the Article submitted to Royal Society M. Barbier, P. Vasilopoulos, and F. M. Peeters transcendental equationscos k x L = cos αL cos P + D α sin P, (3.12a)cos k x L = cos βL cos P + D β sin P, (3.12b)with D λ = (cid:2) ( λ + ε ) cos λL − λ + ε (cid:3) / λ ε , and λ = α, β . To see whether thereis a gap in the spectrum we look for a solution with ε = 0 in the dispersion relations.This gives two values for k x where zero energy solutions occur k x, = ± arccos[1 − ( L /
8) sin P ] /L, (3.13)and the crossing points are at ( ε, k x , k y ) = (0 , ± k x, , k x, value is notreal, then there is no solution at zero energy and a gap arises in the spectrum. FromEq. (3.12a) we see that for sin P > /L a band gap arises. Conductivity.
In bilayer graphene the diffusive dc conductivity, given by Eq. (2.27),takes the form σ µµ ( ε F ) /σ = ( k F / πε F ) (cid:104) ± δ/ k F δ + 1 / / (cid:105) , (3.14)with k F = [ ε F + ∆ ∓ ( ε F δ − ∆ ) / ] / , δ = 1 + 4∆ , and σ = e τ F t ⊥ / (cid:126) . Figure 17. (Color online) Conductivities, σ xx in (a) and σ yy in (b), vs Fermi energy forthe four types of SLs with L = 20 nm and W b = W w = 10 nm, at temperature T = 45 K ; σ = e τ F t ⊥ / (cid:126) . Type I : ∆ b = 50 meV, ∆ w = 25 meV and V b = V w = 0. Type II :∆ b = ∆ w = 25 meV and V b = − V w = 50 meV. Type III : ∆ b = ∆ w = 50 meV and V b = − V w = 25 meV. Type IV : ∆ b = − ∆ w = 100 meV and V b = V w = 0. In Figs. 17(a), (b) the conductivities σ xx in (a) and σ yy in (b) for bilayergraphene are shown for the various types of SLs defined in Sec. 3(b). Notice that fortype IV SL the conductivities σ xx and σ yy differ substantially due to the anisotropyin the spectrum.
4. Conclusions
We reviewed the electronic band structure of single-layer and bilayer graphene inthe presence of 1D periodic potentials. In addition, we investigated the conditionsthat lead to carrier collimation in single-layer graphene and determined when extraDirac points appear in the spectrum and what their influence is on the conductivity.Furthermore, we investigated the tunnelling through, and bound states created by,
Article submitted to Royal Society eview: Single-layer and bilayer graphene SLs k space with k y = 0 andnonzero k y . Type IV SLs tend to split the K (K’) valley into two valleys.In the Kronig-Penney limit, where we take the barriers to be δ functions V ( x ) / (cid:126) v F = P δ ( x ), we saw that the SL spectra, the transmission, conductance, etc., are periodicin the strength of the barriers. As s well known, this is not the case for standardelectrons. An important qualitatively new feature is encountered in the extendedKronig-Penney limit for P = ( n + 1 / π , see Sec. 2(d): the Dirac point becomes aDirac line.We expect that these relatively recent findings, that we reviewed in this work,will be tested experimentally in the near future. This work was supported by IMEC, the Flemish Science Foundation (FWO-Vl), the Bel-gian Science Policy (IAP), and the Canadian NSERC Grant No. OGP0121756.
References
Abergel, D. S. L., Apalkov, V., Berashevich, J., Ziegler, K. & Chakraborty, T. 2010Properties of graphene: A theoretical perspective.Abergel, D. S. L., Apalkov, V., Berashevich, J., Ziegler, K. and Chakraborty,Tapash 2010 Properties of graphene: a theoretical perspective, Advances inPhysics, 59: 4, 261 482Arovas, D. P., Brey, L., Fertig, H. A., Kim, E. & Ziegler, K. 2010 Dirac Spectrumin Piecewise Constant One-Dimensional Potentials.
ArXiv e-prints .Barbier, M., Peeters, F. M., Vasilopoulos, P. & Pereira Jr, J. M. 2008 Dirac andklein-gordon particles in one-dimensional periodic potentials.
Phys. Rev. B , (11), 115 446. (doi:10.1103/PhysRevB.77.115446)Barbier, M., Vasilopoulos, P. & Peeters, F. M. 2009 a Dirac electrons in a kronig-penney potential: Dispersion relation and transmission periodic in the strength ofthe barriers.
Phys. Rev. B , (20), 205 415. (doi:10.1103/PhysRevB.80.205415)Barbier, M., Vasilopoulos, P. & Peeters, F. M. 2010 Extra dirac points in the energyspectrum for superlattices on single-layer graphene. Phys. Rev. B , (7), 075 438.(doi:10.1103/PhysRevB.81.075438)Barbier, M., Vasilopoulos, P., Peeters, F. M. & Pereira Jr, J. M. 2009 b Bilayergraphene with single and multiple electrostatic barriers: Band structure andtransmission.
Phys. Rev. B , (15), 155 402. (doi:10.1103/PhysRevB.79.155402) Article submitted to Royal Society M. Barbier, P. Vasilopoulos, and F. M. Peeters
Bliokh, Y. P., Freilikher, V., Savel’ev, S. & Nori, F. 2009 Transport and localizationin periodic and disordered graphene superlattices.
Phys. Rev. B , (7), 075 123.(doi:10.1103/PhysRevB.79.075123)Brey, L. & Fertig, H. A. 2009 Emerging zero modes for graphene in a periodic po-tential. Phys. Rev. Lett. , (4), 046 809. (doi:10.1103/PhysRevLett.103.046809)Castro, E. V., Novoselov, K. S., Morozov, S. V., Peres, N. M. R., dos Santos, J. M.B. L., Nilsson, J., Guinea, F., Geim, A. K. & Neto, A. H. C. 2007 Biased bilayergraphene: Semiconductor with a gap tunable by the electric field effect. Phys.Rev. Lett. , (21), 216 802. (doi:10.1103/PhysRevLett.99.216802)Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K.2009 The electronic properties of graphene. Rev. Mod. Phys. , (1), 109–162.(doi:10.1103/RevModPhys.81.109)Charbonneau, M., van Vliet, K. M. & Vasilopoulos, P. 1982 Linear response theoryrevisited. III. one-body response formulas and generalized Boltzmann equations. J. Math. Phys. , (2), 318–336.Dragoman, D., Dragoman, M. & Plana, R. 2010 Tunable electrical superlatticesin periodically gated bilayer graphene. J. Appl. Phys. , (4), 044 312. (doi:10.1063/1.3309408)Giovannetti, G., Khomyakov, P. A., Brocks, G., Kelly, P. J. & van den Brink,J. 2007 Substrate-induced band gap in graphene on hexagonal boron nitride:Ab initio density functional calculations. Phys. Rev. B , (7), 073 103. (doi:10.1103/PhysRevB.76.073103)Gomes, J. V. & Peres, N. M. R. 2008 Tunneling of dirac electrons through spatialregions of finite mass. J. Phys.: Condensed Matter , (32), 325 221.Ho, J. H., Chiu, Y. H., Tsai, S. J. & Lin, M. F. 2009 Semimetallic graphene ina modulated electric potential. Phys. Rev. B , (11), 115 427. (doi:10.1103/PhysRevB.79.115427)Huard, B., Sulpizio, J. A., Stander, N., Todd, K., Yang, B. & Gordon, D. G. 2007Transport measurements across a tunable potential barrier in graphene. Phys.Rev. Lett. , (23), 236 803. (doi:10.1103/PhysRevLett.98.236803)Katsnelson, M. I., Novoselov, K. S. & Geim, A. K. 2006 Chiral tunnelling and theklein paradox ingraphene. Nat. Phys. , (9), 620–625. (doi:10.1038/nphys384)Klein, O. 1929 Die reflexion von elektronen an einem potentialsprung nach derrelativistischen dynamik von dirac. Zeitschrift f¨ur Physik A Hadrons and Nuclei , (3), 157–165. (doi:10.1007/BF01339716)Martin, I., Blanter, Y. M. & Morpurgo, A. F. 2008 Topological confinement inbilayer graphene. Phys. Rev. Lett. , (3), 036 804. (doi:10.1103/PhysRevLett.100.036804) Article submitted to Royal Society eview: Single-layer and bilayer graphene SLs
Appl. Phys. Lett. , (21),213106. (doi:10.1063/1.3263150)McCann, E. 2006 Asymmetry gap in the electronic band structure of bilayergraphene. Phys. Rev. B , (16), 161 403. (doi:10.1103/PhysRevB.74.161403)Novoselov, K. S., Geim, A. K., Morozov, S. V., Jiang, D., Zhang, Y., Dubonos,S. V., Grigorieva, I. V. & Firsov, A. A. 2004 Electric field effect in atomicallythin carbon films. Science , (5696), 666–669. (doi:10.1126/science.1102896)Park, C.-H., Son, Y.-W., Yang, L., Cohen, M. L. & Louie, S. G. 2009 a Electronbeam supercollimation in graphene superlattices.
Nano Lett. , (9), 2920–2924.(doi:10.1021/nl801752r)Park, C.-H., Son, Y.-W., Yang, L., Cohen, M. L. & Louie, S. G. 2009 b Landaulevels and quantum hall effect in graphene superlattices.
Phys. Rev. Lett. , (4),046 808. (doi:10.1103/PhysRevLett.103.046808)Park, C.-H., Yang, L., Son, Y.-W., Cohen, M. L. & Louie, S. G. 2008 a Anisotropicbehaviours of massless dirac fermions in graphene under periodic potentials.
Nat.Phys. , (3), 213–217.Park, C.-H., Yang, L., Son, Y.-W., Cohen, M. L. & Louie, S. G. 2008 b New gener-ation of massless dirac fermions in graphene under external periodic potentials.
Phys. Rev. Lett. , (12), 126 804. (doi:10.1103/PhysRevLett.101.126804)Pereira Jr, J. M., Mlinar, V., Peeters, F. M. & Vasilopoulos, P. 2006 Confined statesand direction-dependent transmission in graphene quantum wells. Phys. Rev. B , (4), 045 424. (doi:10.1103/PhysRevB.74.045424)Pereira Jr, J. M., Peeters, F. M., Chaves, A. & Farias, G. A. 2010 Klein tunneling insingle and multiple barriers in graphene. Semiconductor Science and Technology , (3), 033 002. (doi:10.1088/0268-1242/25/3/033002)Pereira Jr, J. M., Vasilopoulos, P. & Peeters, F. M. 2007 a Graphene-based resonant-tunneling structures.
Appl. Phys. Lett. , (13), 132122. (doi:10.1063/1.2717092)Pereira Jr, J. M., Vasilopoulos, P. & Peeters, F. M. 2007 b Tunable quantum dotsin bilayer graphene.
Nano Lett. , (4), 946–949. (doi:10.1021/nl062967s)Roslyak, O., Iurov, A., Gumbs, G. & Huang, D. 2010 Unimpeded tunneling ingraphene nanoribbons. J. Phys.: Condensed Matter , (16), 165 301.San-Jose, P., Prada, E., McCann, E. & Schomerus, H. 2009 Pseudospin valve inbilayer graphene: Towards graphene-based pseudospintronics. Phys. Rev. Lett. , (24), 247 204. (doi:10.1103/PhysRevLett.102.247204)Schliemann, J., Loss, D. & Westervelt, R. M. 2005 Zitterbewegung of electronicwave packets in iii-v zinc-blende semiconductor quantum wells. Phys. Rev. Lett. , (20), 206 801. (doi:10.1103/PhysRevLett.94.206801) Article submitted to Royal Society M. Barbier, P. Vasilopoulos, and F. M. Peeters
Sun, J., Fertig, H. A. & Brey, L. 2010 Effective Magnetic Fields in Graphene Su-perlattices.
ArXiv e-prints .Wang, L.-G. & Zhu, S.-Y. 2010 Electronic band gaps and transport properties ingraphene superlattices with one-dimensional periodic potentials of square barri-ers.
Phys. Rev. B , (20), 205 444. (doi:10.1103/PhysRevB.81.205444)Winkler, R., Z¨ulicke, U. & Bolte, J. 2007 Oscillatory multiband dynamics of freeparticles: The ubiquity of zitterbewegung effects. Phys. Rev. B , (20), 205 314.(doi:10.1103/PhysRevB.75.205314)Young, A. F. & Kim, P. 2009 Quantum interference and klein tunnelling ingraphene heterojunctions. Nat. Phys. , (3), 222–226. (doi:10.1103/PhysRevLett.98.236803)Zawadzki, W. 2005 Zitterbewegung and its effects on electrons in semiconductors. Phys. Rev. B , (8), 085 217. (doi:10.1103/PhysRevB.72.085217)(8), 085 217. (doi:10.1103/PhysRevB.72.085217)