The energy spectra of the graphene-based quasi-periodic superlattice

The spectra of the Dirac quasi-electrons transmission through the Fibonacci quasi-periodical superlattice (SL) are calculated and analyzed in the continuum model with the help of the transfer matrix method. The one-dimensional SL based on a monolayer graphene modulated by the Fermi velocity barriers...

Full description

Saved in:

Bibliographic Details
Published in:	Физика низких температур
Date:	2018
Main Authors:	Korol, A.M., Sokolenko, A.I., Sokolenko, I.V.
Format:	Article
Language:	English
Published:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2018
Subjects:	Наноструктуры при низких температурах
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/176212
Tags:	Add Tag No Tags, Be the first to tag this record!
Journal Title:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:	The energy spectra of the graphene-based quasi-periodic superlattice / A.M. Korol, A.I. Sokolenko, I.V. Sokolenko // Физика низких температур. — 2018. — Т. 44, № 8. — С. 1025-1032. — Бібліогр.: 22 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	nasplib_isofts_kiev_ua-123456789-176212
record_format	dspace
spelling	Korol, A.M. Sokolenko, A.I. Sokolenko, I.V. 2021-02-04T07:34:59Z 2021-02-04T07:34:59Z 2018 The energy spectra of the graphene-based quasi-periodic superlattice / A.M. Korol, A.I. Sokolenko, I.V. Sokolenko // Физика низких температур. — 2018. — Т. 44, № 8. — С. 1025-1032. — Бібліогр.: 22 назв. — англ. 0132-6414 PACS: 73.21.Cd, 73.63.–b https://nasplib.isofts.kiev.ua/handle/123456789/176212 The spectra of the Dirac quasi-electrons transmission through the Fibonacci quasi-periodical superlattice (SL) are calculated and analyzed in the continuum model with the help of the transfer matrix method. The one-dimensional SL based on a monolayer graphene modulated by the Fermi velocity barriers is studied. A new quasi-periodical factor is proposed to be considered. We show that the Fibonacci quasi-periodic modulation in graphene superlattices with the velocity barriers can be effectively realized by virtue of a difference in the velocity barrier values (no additional factor is needed and we keep in mind that not each factor can provide the quasi-periodicity). This fact is true for a case of normal incidence of quasi-electrons on a lattice. In contrast to the case of other types of the graphene SL spectra studied reveal the remarkable property, namely the periodic character over all the energy scale and the transmission coefficient doesn’t tend asymptotically to unity at rather large energies. Both the conductance (using the known Landauer–Buttiker formula) and the Fano factor for the structure considered are also calculated and analyzed. The dependence of spectra on the Fermi velocity magnitude and on the external electrostatic potential as well as on the SL geometrical parameters (width of barriers and quantum wells) is analyzed. Using the quasi-periodical SL one can control the transport properties of the graphene structures in a wide range. The obtained results can be used for applications in the graphene-based electronics. В континуальній моделі методом трансферних матриць розраховано та проаналізовано спектри трансмісії діраківських квазіелектронів крізь квазіперіодичну надгратку (НГ) Фібоначчі. Розглядається одновимірна НГ на основі моношарового графену, модульована бар’єрами швидкості Фермі. Запропоновано використати новий квазіперіодичний фактор. Показано, що квазіперіодична модуляція Фібоначчі в графенових надгратках із бар’єрами швидкості Фермі може бути ефективно реалізована завдяки різниці в значеннях бар’єрів цієї швидкості (додатковий фактор не потрібен, і слід зазначити, що не кожен фактор може забезпечити квазіперіодичну модуляцію). Цей факт справедливий для випадку нормального падіння квазіелектронів на гратку. На відміну від інших типів вивчених спектрів трансмісії в графенових НГ в даному випадку виявляється нетривіальна їх властивість — періодичність по всій шкалі енергії, так що коефіцієнт пропускання не наближається асимптотично до одиниці при достатньо високих енергіях. Розраховано та проаналізовано провідність (з використанням відомої формули Ландауера–Буттікера) та фактор Фано для даної структури. Проаналізовано залежність спектрів від величини швидкості Фермі та від зовнішнього електростатичного потенціалу, а також від геометричних параметрів НГ (ширин бар’єрів і квантових ям). Використовуючи розглянуті квазіперіодичні НГ, можна регулювати транспортні властивості графенових структур в широкому діапазоні їх параметрів. Отримані результати можуть бути використані для застосування в електроніці на основі графену. It’s a pleasure for us to express our gratitude to Dr. S.I. Litvynchuk for the technical assistance. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Наноструктуры при низких температурах The energy spectra of the graphene-based quasi-periodic superlattice Енергетичні спектри квазі-періодичної надгратки на основі графену Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	The energy spectra of the graphene-based quasi-periodic superlattice
spellingShingle	The energy spectra of the graphene-based quasi-periodic superlattice Korol, A.M. Sokolenko, A.I. Sokolenko, I.V. Наноструктуры при низких температурах
title_short	The energy spectra of the graphene-based quasi-periodic superlattice
title_full	The energy spectra of the graphene-based quasi-periodic superlattice
title_fullStr	The energy spectra of the graphene-based quasi-periodic superlattice
title_full_unstemmed	The energy spectra of the graphene-based quasi-periodic superlattice
title_sort	energy spectra of the graphene-based quasi-periodic superlattice
author	Korol, A.M. Sokolenko, A.I. Sokolenko, I.V.
author_facet	Korol, A.M. Sokolenko, A.I. Sokolenko, I.V.
topic	Наноструктуры при низких температурах
topic_facet	Наноструктуры при низких температурах
publishDate	2018
language	English
container_title	Физика низких температур
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
format	Article
title_alt	Енергетичні спектри квазі-періодичної надгратки на основі графену
description	The spectra of the Dirac quasi-electrons transmission through the Fibonacci quasi-periodical superlattice (SL) are calculated and analyzed in the continuum model with the help of the transfer matrix method. The one-dimensional SL based on a monolayer graphene modulated by the Fermi velocity barriers is studied. A new quasi-periodical factor is proposed to be considered. We show that the Fibonacci quasi-periodic modulation in graphene superlattices with the velocity barriers can be effectively realized by virtue of a difference in the velocity barrier values (no additional factor is needed and we keep in mind that not each factor can provide the quasi-periodicity). This fact is true for a case of normal incidence of quasi-electrons on a lattice. In contrast to the case of other types of the graphene SL spectra studied reveal the remarkable property, namely the periodic character over all the energy scale and the transmission coefficient doesn’t tend asymptotically to unity at rather large energies. Both the conductance (using the known Landauer–Buttiker formula) and the Fano factor for the structure considered are also calculated and analyzed. The dependence of spectra on the Fermi velocity magnitude and on the external electrostatic potential as well as on the SL geometrical parameters (width of barriers and quantum wells) is analyzed. Using the quasi-periodical SL one can control the transport properties of the graphene structures in a wide range. The obtained results can be used for applications in the graphene-based electronics. В континуальній моделі методом трансферних матриць розраховано та проаналізовано спектри трансмісії діраківських квазіелектронів крізь квазіперіодичну надгратку (НГ) Фібоначчі. Розглядається одновимірна НГ на основі моношарового графену, модульована бар’єрами швидкості Фермі. Запропоновано використати новий квазіперіодичний фактор. Показано, що квазіперіодична модуляція Фібоначчі в графенових надгратках із бар’єрами швидкості Фермі може бути ефективно реалізована завдяки різниці в значеннях бар’єрів цієї швидкості (додатковий фактор не потрібен, і слід зазначити, що не кожен фактор може забезпечити квазіперіодичну модуляцію). Цей факт справедливий для випадку нормального падіння квазіелектронів на гратку. На відміну від інших типів вивчених спектрів трансмісії в графенових НГ в даному випадку виявляється нетривіальна їх властивість — періодичність по всій шкалі енергії, так що коефіцієнт пропускання не наближається асимптотично до одиниці при достатньо високих енергіях. Розраховано та проаналізовано провідність (з використанням відомої формули Ландауера–Буттікера) та фактор Фано для даної структури. Проаналізовано залежність спектрів від величини швидкості Фермі та від зовнішнього електростатичного потенціалу, а також від геометричних параметрів НГ (ширин бар’єрів і квантових ям). Використовуючи розглянуті квазіперіодичні НГ, можна регулювати транспортні властивості графенових структур в широкому діапазоні їх параметрів. Отримані результати можуть бути використані для застосування в електроніці на основі графену.
issn	0132-6414
url	https://nasplib.isofts.kiev.ua/handle/123456789/176212
citation_txt	The energy spectra of the graphene-based quasi-periodic superlattice / A.M. Korol, A.I. Sokolenko, I.V. Sokolenko // Физика низких температур. — 2018. — Т. 44, № 8. — С. 1025-1032. — Бібліогр.: 22 назв. — англ.
work_keys_str_mv	AT korolam theenergyspectraofthegraphenebasedquasiperiodicsuperlattice AT sokolenkoai theenergyspectraofthegraphenebasedquasiperiodicsuperlattice AT sokolenkoiv theenergyspectraofthegraphenebasedquasiperiodicsuperlattice AT korolam energetičníspektrikvazíperíodičnoínadgratkinaosnovígrafenu AT sokolenkoai energetičníspektrikvazíperíodičnoínadgratkinaosnovígrafenu AT sokolenkoiv energetičníspektrikvazíperíodičnoínadgratkinaosnovígrafenu AT korolam energyspectraofthegraphenebasedquasiperiodicsuperlattice AT sokolenkoai energyspectraofthegraphenebasedquasiperiodicsuperlattice AT sokolenkoiv energyspectraofthegraphenebasedquasiperiodicsuperlattice
first_indexed	2025-11-24T16:10:09Z
last_indexed	2025-11-24T16:10:09Z
_version_	1850484041138044928
fulltext	Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 8, pp. 1025–1032 The energy spectra of the graphene-based quasi-periodic superlattice A.M. Korol1,2, A.I. Sokolenko2, and I.V. Sokolenko3 1Laboratory on Quantum Theory in Linkoping, ISIR, P.O. Box 8017, S-580, Linkoping, Sweden 2National University for Food Technologies, 68, Volodymyrska str., Kyiv 01601, Ukraine E-mail: korolam@ukr.net 3Кrypton Ocean Group, 37/97, Zhylians’ka str., Kyiv 01033, Ukraine Received January 1, 2018, revised March 15, 2018, published online June 27, 2018 The spectra of the Dirac quasi-electrons transmission through the Fibonacci quasi-periodical superlattice (SL) are calculated and analyzed in the continuum model with the help of the transfer matrix method. The one- dimensional SL based on a monolayer graphene modulated by the Fermi velocity barriers is studied. A new qua- si-periodical factor is proposed to be considered. We show that the Fibonacci quasi-periodic modulation in graphene superlattices with the velocity barriers can be effectively realized by virtue of a difference in the ve- locity barrier values (no additional factor is needed and we keep in mind that not each factor can provide the quasi-periodicity). This fact is true for a case of normal incidence of quasi-electrons on a lattice. In contrast to the case of other types of the graphene SL spectra studied reveal the remarkable property, namely the periodic character over all the energy scale and the transmission coefficient doesn’t tend asymptotically to unity at rather large energies. Both the conductance (using the known Landauer–Buttiker formula) and the Fano factor for the structure considered are also calculated and analyzed. The dependence of spectra on the Fermi velocity magni- tude and on the external electrostatic potential as well as on the SL geometrical parameters (width of barriers and quantum wells) is analyzed. Using the quasi-periodical SL one can control the transport properties of the graphene structures in a wide range. The obtained results can be used for applications in the graphene-based electronics. PACS: 73.21.Cd Superlattices; 73.63.–b Electron transport in nanomaterials and structures. Keywords: graphene, Fibonacci superlattice, velocity barriers, transmission spectra. Introduction Graphene and the graphene-based structures draw the great attention of researchers in recent years. It is ex- plained by the unique physical properties of graphene, and also by good prospects of its use in the nanoelectronics (see, e.g., [1–4]). It is convenient to control the behavior of the Weyl–Dirac fermions in graphene by means of the ex- ternal electric and magnetic fields, and a lot of publications are devoted to the corresponding problem for this reason. Recently one more way of controlling the electronic pro- perties of the graphene structures, namely by means of the spatial change of the Fermi velocity was offered [5–10]. Some ways of fabrication of structures in which the Fermi velocity of quasi-particles is spatially dependent value were approved [5,6]. This achievement in the technology opens new opportunities for receiving the nanoelectronic devices with the desirable transport properties. It is known that the solution of this problem can be promoted to the great extent by use of the superlattices. This explains the emergence of a number of publications in which the charge carriers behavior in graphene super- lattices of various types is investigated; these SL include the strictly periodic, the disordered ones, SL with barriers of various nature — electrostatic, magnetic, barriers of Fermi velocity (under which we understand the areas of graphene where quasi-particles have different Fermi velocity, smaller or bigger than in the pristine graphene). In particular, in recent papers [7] and especially [11] the influence of the Fermi velocity barriers on the electronic properties of the strictly periodic graphene superlattices was analyzed. It was shown that it is possible to tune the transmission rates from zero to unity only changing the Fermi velocity, also to control the energy gap value and the amplitudes and location of the resonant peaks in the conductivity. © A.M. Korol, A.I. Sokolenko, and I.V. Sokolenko, 2018 A.M. Korol, A.I. Sokolenko, and I.V. Sokolenko Among the specified works, there are some devoted to the quasi-periodic graphene SL [12–16]. The quasi-perio- dic structures, as known, possess the unusual electronic pro- perties of special interest (see, e.g., [17]), such as self-si- milarity, the Cantor nature of the energy spectrum, etc. Motivated by the circumstances stated above we formu- late the purpose of this work as follows: to study the main features of the transmission spectra of the quasi-periodical graphene-based Fibonacci superlattices with the velocity barriers. We show that using the quasi-periodic superlattices gives the additional possibilities to control the transport properties of the graphene-based structures flexibly. We choose the Fibonacci SL because they are considered as the classical quasi-periodic objects, and the majority of the works associated with research of the quasi-periodic sys- tems deal merely with them. Model and formulae Consider the one-dimensional graphene superlattice in which regions with various values of the Fermi velocity are located along the 0x axis: elements a and b refer to аυ and bυ velocities, respectively. Elements a and b are arranged along SL according to the Fibonacci rule so that, for example, we have for the fourth Fibonacci generation (sequence): s4 = abaab. Generally, between the barriers corresponding to elements a and b, there is a quantum well for which the Fermi velocity is equal to unity as in a pristine graphene: 0wυ = υ . As we consider graphene in which the Fermi velocity is dependent on the spatial coordinate r, i.e., ( )=υ υ r the quasi-particles submit to the massless Weyl–Dirac type equation: ( ) ( ) ( ) ( )i E − ⋅∇ φ = φ σ υ r r υ r r , (1) where ( ; )x y= σ σσ is the Pauli two-dimensional matrix; ( ) ( ) ( );A B Tφ = φ φ  r r r is the two-component spinor, T is the transposing symbol. Introducing an auxiliary spinor ( ) ( ) ( )Φ = φr υ r r one can rewrite equation (1) as follows: ( ) ( ) ( )i E− ⋅∇Φ = Φυ r σ r r . (2) Assume that the external potential consists of the peri- odically repeating rectangular velocity barriers along the axis 0x and potential is constant in each jth barrier. The external electrostatic potential U may also be present and inside each barrier Uj(x) = const (piece-wise constant po- tential). In this case using the translational invariance of the solution over the 0y axis it is possible to receive from the Eq. (2): ( ) 2 , 2 2 ,2 0A B j y A B d k k dx Φ + − Φ = , (3) where indices A, B relate to the graphene sublattices A and B, respectively, ( ) /j j jk E U x = − υ  , measurement units 0 1= υ = are accepted. If we represent the solution for eigenfunctions in the form of the plane waves moving in the direct and opposite direction along an axis 0x, we de- rive ( ) 1 1 e ej jiq x iq x j j j j x a b g g − + −          Φ = +          , (4) where 2 2 j j yq k k= − for 2 2 j yk k> and 2 2 j y jq i k k= − oth- erwise, ( ) ( )/j j y j jg q ik E U± = ± + υ − , the top line in (4) pertains to the sublattice A, the lower one — to the sublattice B. The transfer matrix which associates wave functions in points x and x x+ ∆ reads ( ) ( ) ( ) ( ) cos sin1 cos sin cos j j j j j j j j q x i q x M i q x q x  ∆ − θ ∆  =  θ ∆ ∆ + θ  , (5) where ( )arcsin /j y jk kθ = . Meaning that the Fermi velocity depends only on coor- dinate x, i.e. ( ) ( )xυ = υr , it is possible to receive the boundary matching condition from the continuity equation for the current density as follows: ( ) ( )b bw w bwx x− +υ φ = υ φ , (6) where indexes b and w relate to a barrier and a quantum well, respectively, xbw is the coordinate of the barrier-well interface. The coefficient of transmission of quasi-electrons through the superlattice T(E) is evaluated by means of a transfer matrix method and it is equal to 2Т t= , 0 0 0 22 11 12 21 2cos e ei it R R R R− θ θ θ = + − − , where 0θ is the angle of incidence of the quasi-particles on the lattice and the matrix R is expressed via the product of the matrices jM : 1 N j j R M = =∏ , N is the total number of elements in the SL. Energy ranges for which the coeffi- cient of electron transmission through the lattice is close to unity form the allowed bands while the energy gaps corre- spond to values T << 1. Now we have an opportunity to proceed with analyzing the obtained results. Results and discussion Unlike the energy spectra for the known quasi-periodic superlattices, including the graphene ones (see, e.g., [7,15,16]), the spectra of the graphene-based SL with the velocity barriers are periodic over all the energy scale, and the transmission rate T doesn’t tend asymptotically to unity at rather large energies. For comparison, dependences of log T(E) are given in Fig. 1(a) for the Fibonacci fourth 1026 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 8 The energy spectra of the graphene-based quasi-periodic superlattice generation for SL in which the quasi-periodic modulation is achieved due to different values of the Fermi velocity, and for SL on the basis of the gapped graphene in which the quasi-periodic modulation is due to different values of gaps (calculations are carried out on the basis of our previ- ous work [15], Fig. 1(b)). The values of the parameters are as follows: for the first case w = 10 nm, d = 20 nm, 1aυ = , 2bυ = , for the second case w = d = 10 nm, Δa = 0.08 eV, Δb = 0, where Δ denotes the gap’s width, d and w denotes the barrier and the quantum well width, respectively. All calculations (for all figures of this paper) were carried out for the case of the normal incidence of electrons on the superlattice. (Note that in accordance with the known Lan- dauer–Buttiker formula the electrons with ky = 0 make the main contribution to the conductance). It is seen that a certain periodicity of spectra takes place in the second case (this fact hasn’t been noted in the litera- ture as yet) but the amplitude of peaks (and the correspond- ing gap’s width) decreases with increasing in E, on aver- age. The allowed band width increases on average with E increasing and the coefficient of transmission T eventually approaches to unity. This “wavy damped oscillation” in Fig. 1(b) is associated with such property of the spectra as their self-similarity (e.g., [15]). Note that the narrowing of gaps occurs very rapidly. Parameters for the spectra in Fig. 1 are chosen so as to show that their structure for the graphene SL of different nature may be similar. The differ- ence of two spectra is explained by that the velocity barri- ers are dependent on energy [9]. If we make an analogy between tunneling of quasi-particles in graphene through a rectangular electrostatic barrier and tunneling through a velocity barrier, for the potential of the last it is necessary to write down ( ) bU E E E= − υ , (7) in other words expressions for the transmission coefficient T in the specified cases coincide if the condition (7) is sat- isfied. This formula explains the fact that spectra of T(E) for SL with the velocity barriers are periodic over all the energy scale. It is quite naturally that the expression for the transmission rates comprises the term that directly deter- mines the spectra periodicity (see, e.g., the recent papers [7,19,20]). Note further that the graphene superlattices with the ve- locity barriers are characterized by a rich variety of the energy spectra, and also by their high sensitivity to minor changes in geometrical parameters of a lattice. This state- ment is correct in relation not only to quasi-periodically modulated SL, but to strictly periodic lattices as well and it allows for controlling the energy spectra in a wide range. In the general case, i.e., for arbitrary values of the parame- ter values, the energy spectra demonstrate a set of irregu- larly spaced of allowed and forbidden bands. However for some sets of the parameter values spectra are regular and it is natural to take them for analysis in the first place; exam- ples of such spectra are shown in figures of this paper. (The same conclusion in relation to the strictly periodic SL with the velocity barriers was done in [19,20].) Apparently, depending on the parameters of the prob- lem under consideration spectra may differ from each other significantly; they can reveal the simple form with the small minimal period equal to several energy units, but also they can expose much more complicated pattern of bands with the minimal period of several tens of energy units. Each set of values of parameters provides the origi- nal specter with its own minimal period and substructure. In the minimal period of each specter, there is a point with respect to which the specter is symmetric and besides each specter exhibits a symmetric substructure (e.g., Fig. 1(a)). Let us now consider some concrete energy spectra of the graphene Fibonacci SL modulated by the velocity bar- riers. Figure 2 shows the trace map for the initial Fibonacci generations of the SL in which the quasi-periodic modula- tion is created due to different values of the velocity barri- ers, namely 1aυ = , 2bυ = , d = 10 nm, w = 5 nm, the en- ergy range is selected to be the minimal period equal to 2π eV. The trace map investigated is characterized by the following features. For the taken set of parameters which corresponds to the trace map in Fig. 2 each Fibonacci gen- eration forms spectra with a regular arrangement of the energy bands, and each of them exposes its own geometry. The higher generation is, the spectra of more complex pat- Fig. 1. Dependences of log T on energy E for the SL modulated by: (a) different values of the Fermi velocity and (b) different magni- tudes of the energy gaps. Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 8 1027 0 0 –2 2– –4 –4 –6 –6 –8 –10 2 2 4 4 6 6 8 8 10 10 E, eV E, eV lo g T lo g T (a) (b) A.M. Korol, A.I. Sokolenko, and I.V. Sokolenko tern correspond to it. Note that spectra of higher genera- tions are strongly fragmented (therefore we don’t represent them), and besides fragmentation degree increases signifi- cantly with increase in geometrical SL parameters d, w. With increasing the number of the Fibonacci sequence the number of gaps increases and their total width becomes larger. The fragmentation of the allowed bands in all gen- erations starting from the third one occurs in accordance with the property of the self-similarity. Note also that, for some energy ranges, there are gaps in every Fibonacci se- quence. It should be noted further that in certain fixed energy are- as, the Fibonacci inflation rule is satisfied: 1 2n n nz z z− −= + , where zn is number of bands in the nth Fibonacci genera- tion. The minimal such energy range is shown in Fig. 2. The numbers of the allowed bands in the consequent Fibo- nacci generations for the parameters chosen are 5, 8, 13, 21 for the 2d, 3d, 4th and 5th sequences, respectively. The main conclusion from the spectra presented is as follows: Fibonacci quasi-periodic modulation in graphene superlattices with the velocity barriers can be effectively realized by virtue of a difference in aυ and bυ values, i.e. in value of the velocity barriers (no additional factor is needed). And this fact is true for a case of normal inci- dence of quasi-electrons on a lattice. (Therefore, the state- ment of authors of [14] that in graphene-based SL (in con- trast to other SL), the quasi-periodic modulation can be “manifested only at oblique incidence” of the Weyl–Dirac electrons on a lattice isn't correct. As the results of this work demonstrate (and also the results of the previous works [13,15,16]) the implementation of the quasi-periodic modulation depends on a quasi-periodicity factor, and we see that if this factor is realized either due to different magnitude of the velocity barriers (as in this work), or by virtue of different values of gaps (as in [15,16]), the quasi- periodic modulation takes place not only at inclined inci- dence of quasi-particles on a lattice but also at their normal incidence.) We have shown above that the Fibonacci quasi-periodic modulation in the graphene SL can be created due to dif- ferent Fermi velocity values in the SL barriers. There is another way to form an effective quasi-periodic modula- tion in the SL under consideration and it is due to different values of the electrostatic barriers in different elements of the array while maintaining the velocity the same along the lattice chain. The external electrostatic potential U has a significant impact on the electron transmission and it is convenient to tune the transmission spectra with the help of this potential. Let us first consider briefly the effect of the external potential U on the strictly periodic SL with the ve- locity barriers. Denote the potential in elements a and b as Ua and Ub, respectively; Ua = Ub for the strictly periodic SL. The potential barriers are considered to be the piece wise constant, they are located along the SL chain (0x axis). The changes in the transmission spectra caused by the electro- static potential are illustrated in Fig. 3 and are as follows: 1) a new (additional) gap appears between the two adjacent gaps which exist in the case of U = 0; 2) a shift of all gaps is observed and it depends on the value of U; 3) the gap width depends on U also. These changes are governed by the important property of the spectra — they are periodic with the potential U. For example, for the parameters of Fig. 3, spectra return to their initial state at intervals δU = 0.25n eV, n is an inte- ger, i.e., the additional gap due to the external potential U doesn’t appear. This means that for certain values of U the electrostatic barriers are perfectly transparent for the Di- rac–Weyl quasi-electrons and thus there is a kind of the Klein paradox manifestation in the SL under consideration. (If a bυ = υ = 1 we have T(E) = 1 for all energies and val- ues of U due to the Klein tunneling.) The widening of gaps is accompanied by the narrowing of those gaps which re- late to the SL with the velocity barriers for U = 0. Fig. 2. Trace map for the initial Fibonacci generations, values of the parameters are as follows: d = 10 nm, w = 5 nm, 1aυ = , 2.bυ = Fig. 3. Transmission spectra for the various values of the electro- static potential U, eV: 0 (a), 0.16 (b), 0.36 (c); the other parame- ters: a bυ = υ = 2, d = 10 nm, w = 5 nm. 1028 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 8 1.6 E, eV lo g T 0 0.4 0.8 1.2 –2 –6 –10 (a) E, eV lo g T 0 0.4 0.8 1.2 1.6 2– 6– –10 (b) E, eV lo g T 0 0.4 0.8 1.2 1.6 –2 –6 10– (c) The energy spectra of the graphene-based quasi-periodic superlattice The magnitude of the period oscillations can be found from the following considerations. According to the Bloch theorem we can write ( ) ( )cos 1 2 Tr w ad w M Mβ + =   , (8) d + w is the lattice period. Calculation of the right side of this equation for the case of normal incidence of electrons yields the expression ( ) ( )cos cosd w E U d Ewβ + = − υ±      , (9) a bυ = υ = υ . The last formula yields a value for the period of oscilla- tions in the transmission spectra U n dδ = πυ . (10) This expression determines the dependence of the peri- od Uδ on the SL geometric parameters (it is inversely pro- portional to the barrier width and holds for each value of the quantum well width) and on the Fermi velocity. Note that formula (10) holds well even for a small number of the SL periods. Figure 4 shows a trace map for the SL under considera- tion for the difference 0.08 eVa bU U U∆ = = π− , other pa- rameters as in Fig. 3, the energy interval is chosen to be equal to the minimal period in Fig. 4. In general, its char- acter is similar to that plotted in Fig. 2 but some of its fea- tures must be noted here. This trace map is regular and gaps are wider than for other values of ΔU even if they are larger than that is if the quasi-periodic factor is stronger. This is due to the fact that the spectra for the Fibonacci SL considered preserve the property of the periodicity in the case of Ua ≠ Ub and the factor of the quasi-periodicity is the secondary to the main property of periodicity. For val- ues of ΔU = 0.5n eV the quasi-periodicity doesn’t manifest itself at all and spectra repeat the initial state, i.e., the one for U = 0. The greatest splitting of the allowed bands is observed for values of ΔU slightly higher than n. The trace map is not regular and symmetric for the arbitrary parame- ter values (for the general case when U ≠ 0.25n eV). We see that the trace map in Fig. 4 is divided into two parts by the gap for energy equal to a little more than 0.8 eV (for ΔU chosen). The number of bands is subjected to the Fibonacci inflation rule in every part: for the initial Fibonacci generations we have the sequence of numbers 3, 4, 7, 11… and 1, 2, 3, 5… in the left and right parts, re- spectively, and totally 4, 6, 10, 16… which differs from the case of Fig. 2. Pay particular attention to the broad (lower energy) bands in each Fibonacci generation in Fig. 4. They corre- spond to the so-called additional or superlattice Dirac bands in a periodic lattice [22]. It plays an important role in the controlling of the SL energy spectra since it is robust against the structural disorder. The location of the middle of such a band (mid-gap) ED is determined by the condi- tion [22] 0d wq d q w+ = , (11) which yields ( ) / .DE Ud d w= + υ (12) This equation for the position of the Dirac superlattice gap is well satisfied for a wide range of the parameters involved even for a small number of the SL periods. The Dirac band width depends on the problem parameters and may be less than the width of the other (Bragg) bands (see, e.g., [15,16,18]). Similar Dirac superlattice gaps exist also in the case of the quasi-periodic Fibonacci SL investigated. The mid-gap position of such a gap may be approximately found by the Eq. (13) (for not a large difference between Ua and Ub). Note further that a characteristic feature of the SL Dirac band is that it doesn’t depend on the lattice period d + w, but it is sensitive to the ratio w/d. This is illustrated in Fig. 5 where log T(E) is plotted for the fourth Fibonacci generation with the parameters: υ = 2, Ua = 0.32 eV, Ub = = 0.28 eV, the dashed line in Fig. 5(a) corresponds to values d = 8 nm, w = 6 nm, for the solid line d = 9.6 nm, w = 7.2 nm; for the solid line in Fig. 5(b) d = 6 nm, w = 8 nm, for the dashed line d = 8 nm, w = 6 nm. We have calculated also the values for the structure considered that can be measured in practice namely the conductance and the Fano factor using the known formulae (it is convenient to use the dimensionless conductance, see, e.g., [22] and references therein): ( ) 2 0 , cosyG T E k d π ′ = θ θ∫ , (13) ( ) 2 2 2 2 1 cos cos T T d F T d π −π π −π − θ θ = θ θ ∫ ∫ . (14) Values of G'(E) and F(E) depends in general on the geo- metric parameters of the superlattice, on the number of SL periods, on the external potential and on the Fermi velocity Fig. 4. Trace map for the initial Fibonacci generations of the SL with the parameters: Ua = 0, Ub = 0.25 eV, a bυ = υ = 2, d = = 10 nm, w = 5 nm. Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 8 1029 A.M. Korol, A.I. Sokolenko, and I.V. Sokolenko magnitude in different elements of the lattice. The depend- ence of G' and F on the quasi-electron energy E is present- ed in Figs. 6, 7 for the strictly periodic superlattice with the parameters: d = w = 10 nm, Ua = Ub = 0, the number of the periods n = 8, bυ = 2, 3, 4 for the curves in Figs. 6(a), 6(b), 6(c) and 7(a), 7(b), 7(c), respectively. We see that the dependences of G' and F on E are indeed sensitive to the Fermi velocity value; G'(E) significantly de- creases with increasing of bυ and the Fano factor deviates substantially from the universal value of 1/3 for the most val- ues of the parameters involved being close to unity in those areas of energy which correspond to wide gaps in the T(E) dependence. Note also that we chose the values of the pa- rameters so that the minimum number of the minima in the G'(E) dependence (in one period) is exactly equal to the Fermi velocity values in the barrier regions, namely bυ = 2, 3, 4 for the curves in Figs. 6(a), 6(b), 6(c), respectively; the size of one period becomes larger with the Fermi velocity increasing and it is equal to 2π, 3π, 4π in Figs. 6(a), 6(b), 6(c), respectively (the periodicity of the spectra for the graphene structures with the velocity barriers was analyzed in detail in [19,20]). Values of the Fermi velocity in Figs. 7(a), 7(b), 7(c) are equal to Fυ = 2, 3, 4, respectively. Figure 8 shows the spectra for G'(E) and F(E) for the SL containing the fourth Fibonacci generation with the pa- rameters: d = w = 10 nm, Ua = Ub = 0, bυ = 2. We used only 2 superlattice periods and, interestingly, this is enough for realization of the efficient quasi-periodic modulation. The minima in the conductivity are associated with the maxima in the Fano factor dependence on E. The calculations show that there are regions in the G'(E) and F(E) dependences which correspond to the Dirac super- lattice gaps. These energy areas do not change their posi- tion as the value of the lattice period d + w changes, while the other extremes are shifted on the energy axis. At the same time, the Dirac gap position is sensitive to the ratio d/w. This is evident from Fig. 9 for the SL for the fourth Fibonacci generation with the parameters: d = 9.6 nm, w = 7.2 nm for Figs. 9(a), 9(c) and d = 8 nm, w = 6 nm for Figs. 9(b), 9(d), other parameters are as follows: Fυ = 2, Ua = 0.32 eV, Ub = 0.28 eV. Fig. 5. (Color online) Dependences of log T on energy E for the fourth Fibonacci generation, values of the parameters: υ = 2, Ua = = 0.32 eV, Ub = 0.28 eV, the solid line in (a) corresponds to values d = 9.6 nm, w = 7.2 nm, for the dashed line d = 8 nm, w = 6 nm, for the solid line in (b) d = 6 nm, w = 8 nm, for the dashed line d = 8 nm, w = 6 nm. Fig. 7. Fano factor versus energy for the strictly periodic SL with different Fermi velocity values. Fig. 6. Dependences of the conductance on energy for the strictly periodic SL with different Fermi velocity values υb = 2 (a), 3 (b), 4 (c). 1030 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 8 0 0 –5 –5 –10 –10 E, eV E, eV lo g T lo g T (a) (b) 2 2 4 4 6 6 8 8 10 10 10 0 10 –2 10 –4 G? 0 5 10 15 E, eV 10 0 10 –2 10 –4 G? 0 5 10 15 E, eV 10 0 10 –2 10 –4 G? 0 5 10 15 E, eV (a) (b) (c) 10 0 10 –2 10 –4 0 5 10 15 E, eV 10 0 10 –2 10 –4 0 5 10 15 E, eV 10 0 10 –2 10 –4 0 5 10 15 E, eV (a) (b) (c) F F F The energy spectra of the graphene-based quasi-periodic superlattice Conclusions We analyze the transmission spectra of the Fibonacci superlattice based on graphene modulated by the Fermi ve- locity barriers. The dependences of the transmission rates, of the conductance and of the Fano factor on the quasi- electron energy are calculated and analyzed. The quasi- periodic modulation can be realized due to different values of the velocity barriers or due to different values of the external potential in the SL elements a and b. Contrary to the case of other types of the graphene SL spectra studied reveal the periodic character over all the energy scale and the transmission coefficient doesn’t tend asymptotically to unity at rather large energies. The periodic dependence of the considered spectra on the magnitude of the external electrostatic potential is observed the period being propor- tional to the quantity nπ (n is an integer) and inversely proportional to the barrier width. Spectra demonstrate the rich variety of configurations (patterns) of the allowed and forbidden bands location dependent on one hand on the Fermi velocity magnitude and on the other hand on the SL geometry; for some special parameter values, they expose the regular character, symmetrical with respect to a certain point. Spectra of higher generations are strongly fragment- ed and besides fragmentation degree increases significantly with increase in geometrical SL parameters d, w (the width of the barrier and of the quantum well, respectively). The higher generation is, the spectra of more complex pattern correspond to it. In the certain fixed energy areas the spectra are subjected to the Fibonacci inflation rule: 1 2 ,n n nz z z− −= + where zn is the number of bands in the nth Fibonacci gen- eration. There is another way to form an effective quasi- periodic modulation in the SL under consideration and it is due to different values of the electrostatic barriers in dif- ferent elements of the array while maintaining the velocity the same along the lattice chain. The dependence of the conductance on energy reveals the periodical character and, in particular, one can choose the values of the parame- ters so that the minimum number of the minima in the G'(E) dependence (in one period) is exactly equal to the Fermi velocity values in the barrier regions. The SL Dirac gaps are present in the spectra and their location depends on the velocity barriers value, on the value of the external potential as well as on the SL geometrical parameters. The results of our work can be applied for controlling the ener- gy spectra of the graphene-based devices. It’s a pleasure for us to express our gratitude to Dr. S.I. Litvynchuk for the technical assistance. Fig. 8. Dependences of the conductance (a) and the Fano factor (b) on energy for the SL related to the fourth Fibonacci generation. Fig. 9. Dependences of the conductance (a), (b) and the Fano factor (c), (d) for the SL for the fourth Fibonacci sequence with different values of U, d, w. Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 8 1031 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 5 5 10 10 G? E, eV E, eV (a) (b) F 1.0 1.0 1.0 1.0 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 0 0 0 5 5 5 5 10 10 10 10 E, eV E, eV E, eV E, eV (a) (b) (c) (d) G? G? F F A.M. Korol, A.I. Sokolenko, and I.V. Sokolenko _______ 1. A. Geim and K. Novoselov, Nat. Mater. 6, 183 (2007). 2. A.N. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, Rev. Mod. Phys. 81, 109 (2009). 3. J.M. Pereira, F.M. Peeters, A. Chaves, M. Barbier, and P. Vasilopoulos, Semicond. Sci. Techn. 25, 033002 (2010). 4. V.V. Cheianov and V.I. Falko, Phys. Rev. B 74, 041403 (2006). 5. L. Tapaszto, G. Dobrik, P. Nemes-Incze, G. Vertesy, Ph. Lambin, and L.P. Biro, Phys. Rev. B 78, 33407 (2008). 6. A. Raoux, M. Polini, R. Asgari, A.R. Hamilton, R. Fazio, and A.H. MacDonald, Phys. Rev. B 81, 073407 (2010). 7. P.M. Krstajic and P. Vasilopoulos, J. Phys.: Condens. Matter 23, 135302 (2011). 8. Liu Lei, Li Yu-Xian, and Liu Jian-Jun, Phys. Lett. A 376, 3342 (2012). 9. A. Concha and Z. Tesanovic, Phys. Rev. B 82, 033413 (2010). 10. Y. Wang, Y. Liu, and B. Wang, Physica E 53, 186 (2013). 11. P. Zhao and X. Chen, Appl. Phys. Lett. 99, 182108 (2011). 12. Lu Wei-Tao, Wang Shun-Jin, Wang Jong-Long, Jiang Hua, and Li Wen, Phys. Lett. A 377, 1368 (2013). 13. Xu Yafang, Zou Jianfei, and Jin Guojun, J. Phys.: Condens. Matter 25, 245301 (2013). 14. A.N. Korol and V.N. Isai, Phys. Solid State 55, 2596 (2013). 15. A.Н. Король, ФНТ 40, 324 (2014) [Low Temp. Phys. 40, 251 (2014)]. 16. Z. Cheng, R. Savit, and R. Merlin, Phys. Rev. B 37, 4375 (1988). 17. A.N. Korol and V.N. Isai, Springer Proceegings in Physics 156, 43 (2014). 18. P.V. Ratnikov and A.P. Silin, JETP Lett. 100, 311 (2014). 19. L.-F. Sun, C. Fang, and T.-X. Liang, Chin. Phys. Lett. 30, 047201 (2013). 20. L.-G. Wang and X. Chen, J. Appl. Phys. 109, 033710 (2011). 21. Neetu Agrawal (Garg), Sameer Grover, Sankalpa Ghosh, and Manish Sharma, J. Phys.: Condens. Matter 24, 175003 (2012). 22. Jonas R.F. Lima, Luiz Felipe C. Pereira, and C.G. Bezerra, J. Appl. Phys. 119, 244301 (2016). ___________________________ Енергетичні спектри квазі-періодичної надгратки на основі графену А.М. Король, А.І. Соколенко, І.В. Соколенко В континуальній моделі методом трансферних матриць розраховано та проаналізовано спектри трансмісії діраківських квазіелектронів крізь ква- зіперіодичну надгратку (НГ) Фібоначчі. Розгля- дається одновимірна НГ на основі моношарового графену, модульована бар’єрами швидкості Фер- мі. Запропоновано використати новий квазіперіо- дичний фактор. Показано, що квазіперіодична модуляція Фібоначчі в графенових надгратках із бар’єрами швидкості Фермі може бути ефективно реалізована завдяки різниці в значеннях бар’єрів цієї швидкості (додатковий фактор не потрібен, і слід зазначити, що не кожен фактор може забез- печити квазіперіодичну модуляцію). Цей факт справедливий для випадку нормального падіння квазіелектронів на гратку. На відміну від інших типів вивчених спектрів трансмісії в графенових НГ в даному випадку виявляється нетривіальна їх властивість — періодичність по всій шкалі енер- гії, так що коефіцієнт пропускання не наближа- ється асимптотично до одиниці при достатньо ви- соких енергіях. Розраховано та проаналізовано провідність (з використанням відомої формули Ландауера–Буттікера) та фактор Фано для даної структури. Проаналізовано залежність спектрів від величини швидкості Фермі та від зовнішнього електростатичного потенціалу, а також від геоме- тричних параметрів НГ (ширин бар’єрів і кванто- вих ям). Використовуючи розглянуті квазіперіо- дичні НГ, можна регулювати транспортні властивості графенових структур в широкому ді- апазоні їх параметрів. Отримані результати мо- жуть бути використані для застосування в елект- роніці на основі графену. Ключові слова: графен, надгратка Фібоначчі, бар’єри швидкості Фермі, спектри трансмісії. 1032 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 8 https://doi.org/10.1038/nmat1849 https://doi.org/10.1103/RevModPhys.81.109 https://doi.org/10.1088/0268-1242/25/3/033002 https://doi.org/10.1103/PhysRevB.74.041403 https://doi.org/10.1103/PhysRevB.78.233407 https://doi.org/10.1103/PhysRevB.81.073407 https://doi.org/10.1088/0953-8984/23/13/135302 https://doi.org/10.1088/0953-8984/23/13/135302 https://doi.org/10.1016/j.physleta.2012.08.047 https://doi.org/10.1103/PhysRevB.82.033413 https://doi.org/10.1016/j.physe.2013.05.010 https://doi.org/10.1063/1.3658394 https://doi.org/10.1016/j.physleta.2013.03.035 https://doi.org/10.1088/0953-8984/25/24/245301 https://doi.org/10.1088/0953-8984/25/24/245301 https://doi.org/10.1134/S1063783413120147 https://doi.org/10.1103/PhysRevB.37.4375 https://doi.org/10.1007/978-3-319-06611-0_3 https://doi.org/10.1134/S0021364014170123 https://doi.org/10.1088/0256-307X/30/4/047201 https://doi.org/10.1063/1.3525270 Introduction Model and formulae Results and discussion Conclusions

The energy spectra of the graphene-based quasi-periodic superlattice

Institution

Similar Items