Group velocity in finite graphene superlattices

Dios-Leyva, M. de; Cuan, R.

doi:10.1016/j.spmi.2015.03.046

Cited by 8 publications

(8 citation statements)

References 19 publications

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“…Furthermore, for a N ‐period SL

([], italicBABA . . . italicBA)

consisting of alternating layers of wells A and barriers B , sandwiched between two semi‐infinite layers A of graphene, the dispersion relation () and transmission coefficient are given by

{tan k}_{x} L = \frac{g false(E, k_{y} false) tan N β}{sin β} = F false(E, k_{y} false),

T_{N} false(E, k_{y} false) = \frac{1}{{cos}^{2} N β + g {false(E, k_{y} false)}^{2} \frac{{sin}^{2} N β}{{sin}^{2} β}},

being N the number of unit cells,

g false(E, k_{y} false) = sin {aQ}_{1} cos {bQ}_{2} - h cos {aQ}_{1} sin {bQ}_{2},

and

β = italickd

is the Bloch phase of the corresponding infinite graphene SL, which satisfies the equation:

\begin{matrix} right cos β & center = & left cos {italicaQ}_{1} cos {italicbQ}_{2} + h sin {italicaQ}_{1} sin {italicbQ}_{2} \\ right & center = & left f (E, k_{y}) . \end{matrix}

…”

Section: Transmission Coefficient and Dispersion Relationsmentioning

confidence: 99%

“…determines the energy E as a function of the wavevectors k x and k y , that is, the dispersion relation. Furthermore, for a N-period SL [BABA...BA] consisting of alternating layers of wells A and barriers B, sandwiched between two semi-infinite layers A of graphene, the dispersion relation [8] and transmission coefficient [8,13] are given by…”

Section: Transmission Coefficient and Dispersion Relationsmentioning

confidence: 99%

“…This is achieved by using Eqs. (8) and (5), which, after some algebraic manipulation, lead to the expressions…”

Section: Density Of Statesmentioning

confidence: 99%

“…for the integrand of the infinite and N-period SLs, respectively, with where f = f (E, k y ) and g =g(E, k y ) are shown in Eqs. (8) and (7) and T N (E, k y ) is the transmission coefficient. When substituting Eq.…”

Section: Density Of Statesmentioning

confidence: 99%

“…The latter is clearly illustrated in Ref. [8], where finite-size effects on the dispersion relation and group velocity of the infinite SL were investigated by using the N-period graphene SLs. It is then expected that the properties of the infinite superlattice DOS mentioned above [1,2,4] can be affected by such effects in a very profound manner.…”

mentioning

confidence: 95%

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Graphene superlattices: Effect of finite size on the density of states and conductance

Hernández-Bertrán

Duque

Dios-Leyva

2016

Phys. Status Solidi B

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We have derived a formula for the density of states ρNfalse(Efalse) of a N‐period graphene superlattice (SL), which is given as an integral over the inverse of the absolute value of the group delay velocity vx along the SL‐axis. Using that formula, it was shown that ρNfalse(Efalse) exhibits essentially the same structure for all values of N≥5. It was found that for E<0, the effects of finite crystal size modify dramatically the density of states of the corresponding infinite SL, whereas for E>0 and N≥5, it is only slightly modified. According to our results, 1/vx is proportional to the transmission coefficient, which allows us to establish a certain correlation between the properties of ρNfalse(Efalse) and those of the Landauer conductance GNfalse(Efalse) of the N‐period SL. Certainly, GNfalse(Efalse) exhibits a peak structure as a function of E, with local dips located at the same energies as those of ρNfalse(Efalse). The same behavior was observed for the V‐dependence of GNfalse(Efalse) with E=0, which is very similar to that of ρNfalse(0false). When N increases, the peak positions of both GNfalse(0false) and ρNfalse(0false) tend to be located at those values of V where new Dirac points appear.

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“…Furthermore, for a N ‐period SL

([], italicBABA . . . italicBA)

consisting of alternating layers of wells A and barriers B , sandwiched between two semi‐infinite layers A of graphene, the dispersion relation () and transmission coefficient are given by

{tan k}_{x} L = \frac{g false(E, k_{y} false) tan N β}{sin β} = F false(E, k_{y} false),

T_{N} false(E, k_{y} false) = \frac{1}{{cos}^{2} N β + g {false(E, k_{y} false)}^{2} \frac{{sin}^{2} N β}{{sin}^{2} β}},

being N the number of unit cells,

g false(E, k_{y} false) = sin {aQ}_{1} cos {bQ}_{2} - h cos {aQ}_{1} sin {bQ}_{2},

and

β = italickd

is the Bloch phase of the corresponding infinite graphene SL, which satisfies the equation:

\begin{matrix} right cos β & center = & left cos {italicaQ}_{1} cos {italicbQ}_{2} + h sin {italicaQ}_{1} sin {italicbQ}_{2} \\ right & center = & left f (E, k_{y}) . \end{matrix}

…”

Section: Transmission Coefficient and Dispersion Relationsmentioning

confidence: 99%

Section: Transmission Coefficient and Dispersion Relationsmentioning

confidence: 99%

“…This is achieved by using Eqs. (8) and (5), which, after some algebraic manipulation, lead to the expressions…”

Section: Density Of Statesmentioning

confidence: 99%

Section: Density Of Statesmentioning

confidence: 99%

mentioning

confidence: 95%

See 3 more Smart Citations

Graphene superlattices: Effect of finite size on the density of states and conductance

Hernández-Bertrán

Duque

Dios-Leyva

2016

Phys. Status Solidi B

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Optical Absorption in Periodic Graphene Superlattices: Perpendicular Applied Magnetic Field and Temperature Effects

et al. 2018

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A detailed study of the magneto-optical absorption β( ω) is presented for graphene superlattices (SLs) subjected to a perpendicular magnetic field. For a given temperature, this quantity exhibits a resonant peak structure whose characteristics depend on the magnetic field regime, circular polarization of light and SL barrier height. For the intermediate field regime, we demonstrated that the resonant peak structure of β( ω) is directly correlated to the partial joint density of states. Specifically, the latter exhibits van Hove-like singularities and peaks at energies where β( ω) takes its maximum values. We also investigated the magnetoabsorption in the weak field regime for SLs exhibiting one and extra Dirac points in the absence of the field. It was found that for SLs with only one Dirac point, the absorption spectra consist of resonant peaks satisfying the same circular polarization dependent selection rule as that for pristine graphene, except for one of them. For SLs with extra Dirac points, the resonant peaks arise from transitions between singlet subbands or between doublet subbands and satisfy a circular polarization and peak intensity dependent selection rule. It was also found that the resonant structure of β( ω) can be observed experimentally at room temperature in clean SLs.The ORCID identification number(s) for the author(s) of this article can be found under https://doi.

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Magneto‐Optical Absorption in Graphene Superlattices: Dirac Point Effects

Dios-Leyva

Hernández-Bertrán

Morales

et al. 2017

Physica Rapid Research Ltrs

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We investigate the effects of Dirac points on the magneto-optical absorption of graphene superlattices (SLs) in a weak perpendicular magnetic field. It is shown that the absorption spectrum exhibits a resonant peak structure whose characteristics depend on the properties and number of these points.For SLs with only one Dirac point, the spectrum consists of resonant peaks satisfying the same selection rule as that for monolayer graphene, except for one of these peaks. For SLs with extra Dirac points, the resonant peaks arise from transition between singlet subbands or between doublet subbands and satisfy a circular polarization and peak intensity dependent selection rule.When a magnetic field is applied perpendicular to the twodimensional plane of graphene, electrons in the system exhibit a quantized energy spectrum consisting of highly degenerate and nonequidistant Landau levels (LLs). [1,2] This quantization provides the theoretical basis for the understanding of a number of interesting phenomena, such as the anomalous integer quantum Hall effect [3] and unusual magneto-optical properties. [4][5][6][7][8] Now, if in addition to the magnetic field, a one-dimensional (1D) periodic potential is applied to monolayer graphene, the above mentioned quantization can be substantially modified, with the corresponding modifications of the transport and magneto-optical properties. Certainly, instead of nonequidistant LLs, electrons in graphene exhibit an energy spectrum consisting of magnetic subbands, [9][10][11] whose general characteristics depend on the magnetic field regime. Three regimes were identified that generate different features in these subbands. They correspond to the cases of strong, intermediate, and weak fields, and can be characterized by the ratio d=l B of the period d of the superlattice (SL) potential to the cyclotron radius l B . In the strong field regime, l B ( d and the SL magnetic subbands and states closely resemble those of pristine graphene, whereas in the intermediate one, l B $ d and the subbands are very dispersive. [9][10][11] The most interesting physical situation arises for the weak field regime, where l B ) d. In this case, the magnetic subbands are essentially flat and the electron motion in the system may be considered as quasiclassical. [11] Within the latter approach, the magnetic field does not alter the band structure, it only quantizes the existing one. Consequently, the properties of the Dirac cones (points) [12][13][14] present in the energy spectrum of the SL in the absence of the magnetic field, including their number, can play an important role in determining the characteristics of the magneto-optical properties associated with the low-lying magnetic subbands. The latter was explored in a recent work, [11] where a detailed study of the magnetic subbands, wave functions, and transition strengths allows us to establish the conditions under which the optical transitions between the magnetic subbands are approximately allowed. In this letter, we expand upon these studies and exp...

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Group velocity in finite graphene superlattices

Cited by 8 publications

References 19 publications

Graphene superlattices: Effect of finite size on the density of states and conductance

Graphene superlattices: Effect of finite size on the density of states and conductance

Optical Absorption in Periodic Graphene Superlattices: Perpendicular Applied Magnetic Field and Temperature Effects

Magneto‐Optical Absorption in Graphene Superlattices: Dirac Point Effects

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