Quasienergy spectrum of electrons in ac-driven graphene superlattice

Accepted Manuscript Quasienergy spectrum of electrons in ac-driven graphene superlattice

S.V. Kryuchkov, E.I. Kukhar’, E.S. Ionkina PII:

S0749-6036(16)30470-0

DOI:

10.1016/j.spmi.2016.10.026

Reference:

YSPMI 4571

To appear in:

Superlattices and Microstructures

Received Date:

13 July 2016

Revised Date:

09 October 2016

Accepted Date:

12 October 2016

Please cite this article as: S.V. Kryuchkov, E.I. Kukhar’, E.S. Ionkina, Quasienergy spectrum of electrons in ac-driven graphene superlattice, Superlattices and Microstructures (2016), doi: 10.1016 /j.spmi.2016.10.026

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ACCEPTED MANUSCRIPT Highlights

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Quasienergy spectrum of electrons in ac-driven graphene superlattice is derived. Vector of ac-field is assumed to oscillate perpendicular to superlattice axis. Gap between valence and conduction bands changes with ac-field amplitude changing. Miniband width changes with ac-field amplitude changing.

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   

ACCEPTED MANUSCRIPT Quasienergy spectrum of electrons in ac-driven graphene superlattice

S.V. Kryuchkov a,b, E.I. Kukhar’ a *, E.S. Ionkinab

Volgograd State Socio-Pedagogical University, Physical Laboratory of Low-Dimensional

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a

Systems, V.I. Lenin Avenue, 27, Volgograd 400066, Russia b

Volgograd State Technical University, V.I. Lenin Avenue, 28, Volgograd 400005, Russia

Quasienergy spectrum of electrons in graphene superlattice subjected to the ac electric field was investigated. The ac-field vector was assumed to oscillate perpendicularly to the superlattice axis. Minibad width was found to change with ac-

spectrum of graphene superlattice.

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field amplitude changing. Such effect is due to the nonadditivity of the electron

1. Introduction

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Keywords: superlattice; graphene superlattice; Floquet spectrum; quasienergy

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Development of micro- and nanoelectronics is closely related with the creation of new materials. So it stimulates the investigations of optical and electric properties of graphene based

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structures [1]. Modern technologies allow creating low-dimensional structures of any type, and the accuracy can reach up to single atomic layer [2,3]. Currently the study of the electronic properties of the graphene structures has both fundamental [4-8] and practical significance [9-12]. The high mobility of graphene charge carriers at room temperature allows the use of such material for creation of micrometer devices working in the ballistic regime [13]. A number of electronic devices Corresponding author. Tel.: +79270646789. E-mail address: [email protected] (E.I. Kukhar). *

ACCEPTED MANUSCRIPT based on the graphene materials (particularly – samples of field-effect transistors) has been already worked out [14-16]. In Refs. [12,17] the use of graphene for generation of electromagnetic (EM) radiation in THz range had been offered. High-frequency EM radiation is known to lead to the dynamical renormalization of the

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electron spectrum of low-dimensional systems [18]. Induced by electron-photon coupling modification of band structure of graphene based materials is one of the subjects of intense activity last time [19,20]. Some materials (including graphene) can be tuned by time-dependent driving (further it is called as ac-driving) into specific states called as the Floquet topological insulators [21]. Theory describing magneto-electronic properties of ac-driven graphene in quantizing magnetic field had been developed in Refs. [22-24]. Moreover, there were the experiments [25,26] which had shown the possibility of Floquet topological insulators. Such states had been realized in laboratory

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[25]. Also polarization tunable gaps in band structure of topological insulator had been observed in Ref. [26] by using of mid-infrared EM radiation.

Results of investigations of band structure of ac-driven graphene allow the manipulation of

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Dirac points [27] and Landau levels [22-24] by changing of ac-field amplitude. These give the way of spectrum parameters controlling without interfering in the internal structure of a graphene

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materials. The latter is important for the engineering of graphene based devices with tunable characteristics [11]. The high mobility of electrons in graphene, on the one hand, and the possibility

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of using of superlattices (SL) as a working medium of amplifiers of EM radiation and sources of solitary EM waves [28,29], on the other hand, make graphene SL (GSL) attractive to researchers

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[30,31]. Investigations of ac-field modification of band structure are of interest not only for graphene but for GSL also. Thus there is the problem of the ac electric field effect on the electron spectrum of GSL. Below we consider this question and find the quasienergy spectrum of ac-driven GSL within non-resonant approximation.

ACCEPTED MANUSCRIPT 2. AC-driven GSL. Non-resonant approximation A number of investigations had proposed various ways of GSL fabrication. GSL obtained with lines of pairs of adsorbed hydrogen atoms on the graphene layer was suggested in Ref. [32]. In Ref. [33] the additional periodic potential due to the ripples on the graphene sheet was studied. In

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Ref. [34] the GSL was the 2D structure consisted of periodically arranged strips of the graphene between which there were the strips of h-BN. The dispersion law of GSL was investigated in Refs. [35-37] analytically within the Kronig-Penney model. In Ref. [35] GSL was formed by the periodical electrostatic potential applied in the graphene plane. In Ref. [38] GSL was proposed to be obtained by spatial periodic modulation of the Fermi velocity.

We consider the GSL obtained by depositing of graphene on the periodical substrate as well as in Ref. [37]. To this end one can use the substrate h-BN/SiO2 or h-BN/SiC. The presence of such periodical structures yields the spatial periodic modulation of the band gap in the graphene [37]. Let

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profile of such modulation is defined by the function  s  x    s  x  d  , where d is the GSL period and graphene layer is supposed to coincide with plane xy . Besides, GSL is suggested to be

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subjected to the ac electric field with intensity oscillating in the plane xy perpendicularly to the GSL axis so that its vector potential is

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E   A ac   0,  0 sin t  ,   

(1)

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where E0 and  are the amplitude and the frequency of the intensity of the ac-field correspondingly. This can be achieved by placing of the GSL in the field of EM wave which

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propagates perpendicular to the graphene plane xy so that vector of electric field oscillates along the axis Oy . The quantum mechanical state of the electron is described by a spinor  which in the vicinity of Dirac point K obeys the Dirac-like equation which has the form i  t   Hˆ r, t  . Here

Hˆ r, t    Fσˆ  pˆ  e Fˆ y Ayac t    s  x ˆ z  Vdc  x  ,

(2)

ACCEPTED MANUSCRIPT spinor  has two components corresponding to the different values of pseudospin denoting the graphene sublattice, σˆ  ˆ x , ˆ y  , ˆ z are the Pauli matrixes,  F is the Fermi velocity, Vdc  x  is the additional scalar potential periodical along the axis Ox : Vdc  x  d   Vdc  x  . Spinor  obeys the Floquet theorem:

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 r, t   u r, t e  i t , eff

(3)

where  eff is quasienergy, u r, t  is the spinor with components which are time periodic functions with period 2  . After substitution of Eq. (3) into Eq. (2) we have Hˆ ac r, t u   eff u , where

Hˆ ac r, t   i  t   Fσˆ  pˆ  e Fˆ y Ayac t    s  x ˆ z  Vdc  x  .

(4)

To reduce the time-dependent problem described by the Hamiltonian (4) to a stationary problem the spinor u is written in the form of the next linear combination

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u r, t   b  r, t    b  r, t   , where spinors  are equal

(6)

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1  i   i a0 cos t  1 i       e . 2  2 

(5)

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Here a0  e F E0  2 , ˆ z    , ˆ z     . It is easy to show that

 x   i  e 2i a

0

cos t

,  y     ,  z     e 2i a0 cos t .

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Orthonormal functions (6) define the complete system of basis states of electron in ac-driven GSL. Using such system and linear combination (5) we arrive at the next equations

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 i  t b    F pˆ y b   i Fe 2i a0 cos t pˆ xb   e 2i a0 cos t  s  x b    eff  Vdc b 

(7)

Now we use the Fourier series representation of the components of the spinors b  : b  r, t    cn r  eint . n

Then instead of the system (7) we obtain the system of algebraic equations:

n  

F

pˆ y   eff  Vdc cn   i k n J k n 2a0  i F pˆ x   s  ck , k

(8)

ACCEPTED MANUSCRIPT where J n  x  is the Bessel function of integer order. Further we take into account the non-resonant conditions. Namely the next inequalities are supposed to be performed:  c(1)     g ,  g(1) . Here

 c(1) is the width of the first conduction miniband,  g is the band gap width and  g(1) is the width of

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the gap between the first two conduction minibands. Thus the quanta of such EM field can be neither absorbed nor emitted by an electron in GSL. Hence the amplitudes cn are negligibly small if n  0 . Moreover the parameter 2a0 is supposed to be far from the zeros of the Bessel function: J n 2a0 cn  J 0 2a0 c0 , if n  0 . Said above conditions allow to leave in system (8) the terms

with n  k  0 . As a result the problem of the electron state in ac-driven GSL is reduced to the solving of the equation Hˆ 0 0   eff 0 , where we define

 c 

(9)

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Hˆ 0  uFˆ y pˆ x   Fˆ z pˆ y  acs  x ˆ x  Vdc ,

 0   0  , uF   F J 0 2a0  , acs x   J 0 2a0  s x  .  c0 

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After use of the next unitary transformation:

1 Hˆ  Uˆ  Hˆ 0Uˆ ,   Uˆ  0 , Uˆ  ˆ x  ˆ y ˆ x  ˆ z  , 2

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we obtain instead of Eq. (9) the next transformed Hamiltonian (10)

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Hˆ  uFˆ x pˆ x   Fˆ y pˆ y  acs  x ˆ z  Vdc  x  .

Thus in non-resonant approximation the effective Hamiltonian (10) of ac-driven GSL corresponds to the Dirac electron with anisotropic Fermi velocity moving through the periodically arranged

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barriers with profile renormalized by the ac-field action and represented by the function acs  x  .

3. Kronig-Penney model Now we consider the functions  s  x  and Vdc  x  written in the form of square-wave barriers as it is shown in Fig. 1. The widths of well and barrier are defined as d w and d b correspondingly,

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d  dw  db .

The

equation

Hˆ  r    eff  r 

admits

solutions

in

the

form

 r    nw ,b x expip y y  . Here Hˆ is the effective Hamiltonian (11), functions acs x  and Vdc x  are acs  x   J 0 2a0 1 and Vdc  x   0 , if n  1d  x  d w  n  1d ,

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acs  x   J 0 2a0  2 and Vdc  x   V0 , if d w  n  1d  x  nd . The parts of the wave functions defined as  nw , b  x  correspond to the electron states in the well and barrier with number n and satisfy the boundary conditions

 nw d w  0   nb d w  0 ,  nb d  0   nw1  0 .

Dispersion relation in this case is derived from the equation Re T11  cos p x d  , where T11 is the transfer matrix element located in the first row and first column. Transfer matrix can be written as

ˆ 1 0 ˆ d  ˆ 1 d  ˆ d  , Tˆ   1 2 2 w 1 w

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where

1  iˆ z p 1, 2 x u F p 2  i F p y u p  i F p y  , 1  F 1 , 2  , * e  1, 2   eff  J 0 2a0 1  eff  V0  J 0 2a0  2

1 ˆ  x     1, 2  1, 2 

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2 uF2 p12   eff   F2 p y2  J 02 2a0 21 , uF2 p22   eff  V0    F2 p y2  J 02 2a0 22 .

2





uF2 p12  uF2 p22  V02  J 02 2a0  22  21 sin  p1d w sin  p2 d b   2uF2 p1 p2

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

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Having found the real part of T11 we obtain

 cos p1d w  cos p2 d b   cos p x d  .

(11)

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If ac-field is absent ( a0  0 ) and 1   2  0 formula (11) yields the result [35,36]. Also if a0  0 and 1  0 it gives the dispersion relation [37]. The numerical analysis shows that in the presence of ac-field the widths of the dynamically modified gaps and conduction minibands change with amplitude a0 . The dependences of the dynamically modified gap between valence and conduction minibands  g* and of the width of the first conduction miniband  c* on the amplitude a0 are shown in Figs. 2 and 3 for the next

ACCEPTED MANUSCRIPT parameters values: 21  0.053 eV, 2 2  0.26 eV (such values are typical for substrate h-BN/SiC [39,40]),  F  108 cm/s, d  2·10–6 cm, d b d w  4, V0  0 .

4. Tight-binding approximation

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In this section we obtain the quasienergy eigenvalues of the equation Hˆ    eff  with use of tight-binding approximation as well as in Ref. [41]. Here operator Hˆ is Hamiltonian (10) written in the case of absence of scalar potential ( V0  0 ). Conditions of tight-binding approximation lead to the analytical expression for the electron quasienergy spectrum

 eff   F2 p y2  J 02 2a0  2   2 cos p x d  ,

(12)

where parameters  and  are defined by the shape of SL barriers and by the ratio between the

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barrier width and the well width. For square barriers with d b d w  4 and for the pointed above values of 1 ,  2 , and d calculations give  ~ 0.122 eV,  ~ 0.047 eV. If a0  0 then the energy spectrum [41] follows from the formula (12). Thus quasienergy (as well as energy) spectrum is seen

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from Eq. (12) to be nonadditive unlike the conventional semiconductor SL [28]. This fact leads to the mutual influence of the dynamics of the electrons moving through the orthogonal directions.

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The use of the explicit form of the quasienergy spectrum of GSL (12) allows us to find the

GSL ( p y  0 ):

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analytical expressions for the field-modified gap  g* and conduction miniband width  c* of ac-driven

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 g*  2 J 0 2a0   2   2 ,  c*  J 0 2a0 

   2

2



 2   2 .

(13)

Plotted by the formulas (13) dependences of the parameters  g* and  c* on the amplitude a0 are represented in Figs. 2 and 3 by dashed lines. A good agreement between the numerical analysis performed with the help of dispersion relation (11) and analytical model (12) derived with the use of tight-binding approximation is seen from these figures. To decrease the numerical difference between considered models it is necessary to increase the height of the barriers  2 . It can be

ACCEPTED MANUSCRIPT reached by the use of substrate material which induces in the graphene band structure the gap lager than that of graphene on the SiC-substrate. The field-modified gap and conduction miniband width are seen from the Figs. 2 and 3 to change with amplitude a0 . Note that dynamical narrowing of the conduction miniband of

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conventional semiconductor SL with additive spectrum induced by the high-frequency electric field had been studied in Ref. [18], where electric field vector had been assumed to oscillate parallel to the SL axis. But above we have shown that nonadditivity of electron spectrum of GSL leads to the dynamical modification of its band structure even if vector of ac electric field is applied perpendicularly to the GSL axis. Observation of such feature of the quasienergy spectrum is possible in the real fields [25,26], and the problem of the field-induced modification of the electron spectrum of graphene materials is of interest in connection with the possibilities of band parameters

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controlling.

5. Modification of the negative differential conductivity

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The study of the negative differential conductivity (NDC) is the traditional problem of the investigations of electron transport in structures with SL [42,43]. The interest of researchers to such

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investigations is explained by the possibility of creation of generator of Bloch THz generator. One of the causes of NDC is the Bragg reflections at the SL miniband boundaries [42]. The appearance

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of this effect can be easy explained by next reasoning. Let SL is assumed to be subjected by dc electric field with vector Edc applied through the SL axis Ox . If in dc-field electrons of SL acquire

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the energy enough to reach the top of the conduction miniband then the intensity Edc satisfies the condition j x Edc ~  c n0  . Here j x is the current density induced by the dc-field,  c is the miniband width,  is the average time between electron collisions, n0 is the charge carriers concentration. It is seen from the last condition that the increasing of the dc-field intensity Edc leads to the

ACCEPTED MANUSCRIPT decreasing of the current density j x . Note that this conclusion is quite correct while the oneminiband model works and it is true that eEdc d   1 . In the case when SL (or GSL) is additionally subjected to the ac electric field we have to

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make the substitution  c   c* for the condition of NDC appearance. Here  c* is the field-modified miniband width. To find the current density corresponding to NDC we use the model (12). After using the second formula (13) for the parameter  c* we write

jx ~

n0 2 J 0 2a0  .   Edc

In last expression we have taken into account the inequality  2   2 . Thus the slope of the current-voltage characteristic depends on the ac-field amplitude. Similar effect was obtained in Ref. [28] for the semiconductors SL with additive spectrum in the case when high-frequency electric

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field vector oscillated parallel to the SL axis. But here we have shown that ac electric field effects on the NDC of GSL even if its intensity vector oscillates perpendicularly to the GSL axis unlike SL

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with additive spectrum.

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6. Discussion

Above the quasienergy spectrum of electrons in GSL subjected to the ac electric field has

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been calculated. The ac-field has been assumed to have nonresonant character. The calculations have been performed both within the Kronig-Penney model and within the tight-binding

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approximation. In the case when V0  0 the explicit form of the quasienergy spectrum has been derived (formula (12)).

Nonadditivity of GSL electron spectrum has been shown to lead to the dynamical modification of the band gap and the miniband width even if the vector of ac electric field is applied perpendicularly to the GSL axis. Such feature makes GSL different from the conventional SL with additive spectrum where the effect of ac-field with such polarization leads only to a shift of spectrum as a whole. For instance, unlike SL with additive spectrum dynamical effect of EM

ACCEPTED MANUSCRIPT radiation polarized perpendicularly to the GSL axis leads to the dependence of the slope of the GSL current-voltage characteristic on the ac-field amplitude. This is the advantage of GSL over the conventional SL [18,28,29,42,43] where the current-voltage characteristic modification caused by the ac pump field is sensitive to the field polarization and, in particular, it disappears when the

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pump field is polarized perpendicularly to the SL axis.

Dependences of field-modified gap and miniband width in GSL bad structure on the amplitude a0 calculated within the Kronig-Penney model and with the help of formulas (13) derived in the tight-binding approximation are in quite good agreement. Difference between the results obtained within such models can be decreased by increasing of the height  2 of the SL barriers.

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Acknowledgements

The work was supported with the funding of the Ministry of Education and Science of the Russian Federation within the base part of the State task № 2014/411 (Project Code: 3154), and the

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ACCEPTED MANUSCRIPT [29] A.V. Shorokhov, M.A. Pyataev, N.N. Khvastunov, T. Hyart, F.V. Kusmartsev, K.N. Alekseev, JETP Lett. 100 (2015) 766-770. [30] R.V. Gorbachev, J.C.W. Song, G.L. Yu, A.V. Kretinin, F. Withers, Y. Cao, A. Mishchenko, I.V. Grigorieva, K.S. Novoselov, L.S. Levitov, A.K. Geim, Science 346 (2014) 448-451.

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[31] H. Cheng, C. Li, T. Ma, L.-G. Wang, Y. Song, H.-Q. Lin, Appl. Phys. Lett. 105 (2014)

[32] L.A. Chernozatonskii, P.B. Sorokin, E.E. Belova, I. Bryuning, A.S. Fedorov, JETP Lett. 85 (2007) 77-81.

[33] A.L. Vazquez de Parga, F. Calleja, B. Borca, M.C.G. Passeggi, J.J. Hinarejos, F. Guinea, R. Miranda, Phys. Rev. Lett. 100 (2008) 056807.

[34] H. Sevinçli, M. Topsakal, S. Ciraci, Phys. Rev. B 78 (2008) 245402.

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[35] M. Barbier, F.M. Peeters, P. Vasilopoulos, J. Milton Pereira Jr., Phys. Rev. B 77 (2008) 115446.

[36] M. Barbier, P. Vasilopoulos, F.M. Peeters, Phys. Rev. B 81 (2010) 075438.

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[37] P.V. Ratnikov, JETP Lett. 90 (2009) 469-474.

[38] P.V. Ratnikov, A.P. Silin, JETP Lett. 100 (2014) 311-318.

(2007) 073103.

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[39] G. Giovannetti, P.A. Khomyakov, G. Brocks, P.J. Kelly, J. van den Brink, Phys. Rev. B 76

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[40] A. Mattausch, O. Pankratov, Phys. Rev. Lett. 99 (2007) 076802. [41] S.V. Kryuchkov, E.I. Kukhar’, Physica E 46 (2012) 25-29.

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[42] Yu.A. Romanov, Yu.Yu. Romanova, Semiconductors 39 (2005) 147-155. [43] T. Hyart, N.V. Alexeeva, J. Mattas, K.N. Alekseev, Microelectron. J. 40 (2009) 719-721.

ACCEPTED MANUSCRIPT Figure captions

Fig. 1. Spatial periodic modulation of the band gap in graphene

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Fig. 2. Dependences of the field-modified gap between valence and conduction minibands  g* on the ac-field amplitude. Solid line corresponds to the Kronig-Penney model, dashed line corresponds to the tight-binding approximation

Fig. 3. Dependences of the field-modified width of the first conduction miniband  c* on the ac-field amplitude. Solid line corresponds to the Kronig-Penney model, dashed line corresponds to the tight-

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binding approximation

ACCEPTED MANUSCRIPT Quasienergy spectrum of electrons in ac-driven graphene superlattice

S.V. Kryuchkov a,b, E.I. Kukhar’ a *, E.S. Ionkinab

Volgograd State Socio-Pedagogical University, Physical Laboratory of Low-Dimensional

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a

Systems, V.I. Lenin Avenue, 27, Volgograd 400066, Russia b

Volgograd State Technical University, V.I. Lenin Avenue, 28, Volgograd 400005, Russia

Quasienergy spectrum of electrons in graphene superlattice subjected to the ac electric field was investigated. The ac-field vector was assumed to oscillate perpendicularly to the superlattice axis. Minibad width was found to change with ac-

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field amplitude changing. In the vicinity of extra Dirac points of superlattice band structure the gap was shown to be dynamically induced by the ac-field. Such effects are

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due to the nonadditivity of the electron spectrum of graphene superlattice.

1. Introduction

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Keywords: superlattice; graphene superlattice; extra Dirac points; quasienergy

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Development of micro- and nanoelectronics is closely related with the creation of new materials. So it stimulates the investigations of optical and electric properties of graphene based

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structures [1]. Modern technologies allow creating low-dimensional structures of any type, and the accuracy can reach up to single atomic layer [2,3]. Currently the study of the electronic properties of the graphene structures has both fundamental [4-8] and practical significance [9-12]. The high mobility of graphene charge carriers at room temperature allows the use of such material to create micrometer devices working in the ballistic regime [13]. A number of electronic devices based on Corresponding author. Tel.: +79270646789. E-mail address: [email protected] (E.I. Kukhar). *

ACCEPTED MANUSCRIPT the graphene materials (particularly – samples of field-effect transistors) has been already worked out [14-16]. In Refs. [12,17] the use of graphene for generation of electromagnetic (EM) radiation in THz range had been offered. High-frequency EM radiation is known to lead to the dynamical renormalization of the

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electron spectrum of low-dimensional systems [18]. Induced by electron-photon coupling modification of band structure of graphene based materials is one of the subjects of intense activity last time [19-22]. Some materials (including graphene) can be tuned by time-dependent driving (further it is called as ac-driving) into specific states called as the Floquet topological insulators [23]. Laser-induced gap in the vicinity of the Dirac points of the band structure of the originally gapless graphene layer had been predicted in Refs. [19,20]. Theory describing magneto-electronic properties of ac-driven graphene developed in Refs. [24-26]. Moreover, there are the experiments

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[27,28] which showed the possibility of Floquet topological insulators. Such states had been realized in laboratory [27]. Also polarization tunable gaps in band structure of topological insulator had been observed in Ref. [28] by using of mid-infrared EM radiation.

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Results of investigations of band structure of ac-driven graphene allow the manipulation of Dirac points [29] and Landau levels [24-26] by changing of ac-field amplitude. These give the way

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of spectrum parameters controlling without interfering in the internal structure of a graphene materials. The latter is important for the engineering of graphene based devices with tunable

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characteristics [11]. The high mobility of electrons in graphene, on the one hand, and the possibility of using of superlattices (SL) as a working medium of amplifiers of EM radiation and sources of

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solitary EM waves [30,31], on the other hand, make graphene SL (GSL) attractive to researchers [32,33]. And investigations of ac-field modification of band structure are of interest not only for graphene but for GSL also. Thus there is the problem of the ac electric field effect on the electron spectrum of GSL. Below we consider this question, find the quasienergy spectrum of ac-driven GSL within non-resonant approximation and predict the ac-field induced gaps in the vicinity of socalled extra Dirac points in band structure of GSL.

ACCEPTED MANUSCRIPT 2. AC-driven GSL. Non-resonant approximation A number of investigations had proposed various ways of GSL fabrication. GSL obtained with lines of pairs of adsorbed hydrogen atoms on the graphene layer was suggested in Ref. [34]. In Ref. [35] the additional periodic potential due to the ripples on the graphene sheet was studied. In

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Ref. [36] the GSL was the 2D structure consisted of periodically arranged strips of the graphene between which there were the strips of h-BN. The dispersion law of GSL was investigated in Refs. [37-39] analytically within the Kronig-Penney model. In Ref. [37] GSL was formed by the periodical electrostatic potential applied in the graphene plane. In Ref. [40] GSL was proposed to be obtained by spatial periodic modulation of the Fermi velocity.

We consider the GSL obtained by depositing of graphene on the periodical substrate as well as in Ref. [39]. To this end one can use the substrate h-BN/SiO2 or h-BN/SiC. The presence of such periodical structures yields the spatial periodic modulation of the band gap in the graphene [39]. Let

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profile of such modulation is defined by the function  s  x    s  x  d  , where d is the GSL period and graphene layer is supposed to coincide with plane xy . Besides, GSL is suggested to be

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subjected to the ac electric field with intensity oscillating in the plane xy perpendicularly to the GSL axis so that its vector potential is

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E   A ac   0,  0 sin t  ,   

(1)

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where E0 and  are the amplitude and the frequency of the intensity of the ac-field correspondingly. This can be achieved by placing of the GSL in the field of EM wave which

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propagates perpendicular to the graphene plane xy so that vector of electric field oscillates along the axis Oy . The quantum mechanical state of the electron is described by a spinor  which in the vicinity of Dirac point K obeys the Dirac-like equation which has the form i  t   Hˆ r, t  . Here

Hˆ r, t    Fσˆ  pˆ  e Fˆ y Ayac t    s  x ˆ z  Vdc  x  ,

(2)

ACCEPTED MANUSCRIPT spinor  has two components corresponding to the different values of pseudospin denoting the graphene sublattice, σˆ  ˆ x , ˆ y  , ˆ z are the Pauli matrixes,  F is the Fermi velocity, Vdc  x  is the additional scalar potential periodical along the axis Ox : Vdc  x  d   Vdc  x  . Spinor  obeys the Floquet theorem:

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 r, t   u r, t e  i t , eff

(3)

where  eff is quasienergy, u r, t  is the spinor with components which are time periodic functions with period 2  . After substitution of Eq. (3) into Eq. (2) we have Hˆ ac r, t u   eff u , where

Hˆ ac r, t   i  t   Fσˆ  pˆ  e Fˆ y Ayac t    s  x ˆ z  Vdc  x  .

(4)

To reduce the time-dependent problem described by the Hamiltonian (4) to a stationary problem the spinor u is written in the form of the next linear combination

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u r, t   b  r, t    b  r, t   , where spinors  are equal

(6)

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1  i   i a0 cos t  1 i       e . 2  2 

(5)

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Here a0  e F E0  2 , ˆ z    , ˆ z     . It is easy to show that

 x   i  e 2i a

0

cos t

,  y     ,  z     e 2i a0 cos t .

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Orthonormal functions (6) defines the complete system of basis states of electron in ac-driven GSL. Having used such system and linear combination (5) we arrive at the next equations

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 i  t b    F pˆ y b   i Fe 2i a0 cos t pˆ xb   e 2i a0 cos t  s  x b    eff  Vdc b 

(7)

Now we use the Fourier series representation of the components of the spinors b  : b  r, t    cn r  eint . n

Then instead of the system (7) we obtain the system of algebraic equations:

n  

F

pˆ y   eff  Vdc cn   i k n J k n 2a0  i F pˆ x   s  ck , k

(8)

ACCEPTED MANUSCRIPT where J n  x  is the Bessel function of integer order. Further we take into account the non-resonant conditions. Namely the next inequalities are supposed to be performed:  c(1)     g ,  g(1) . Here

 c(1) is the width of the first conduction miniband,  g is the band gap width and  g(1) is the width of

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the gap between the first two conduction minibands. Thus the quanta of such EM field can be neither absorbed nor emitted by an electron in GSL. Hence the amplitudes cn are negligibly small if n  0 . Moreover the parameter 2a0 is supposed to be far from the zeros of the Bessel function: J n 2a0 cn  J 0 2a0 c0 , if n  0 . Said above conditions allow to leave in system (8) the terms

with n  k  0 . As a result the problem of the electron state in ac-driven GSL is reduced to the solving of the equation Hˆ 0 0   eff 0 , where we define

 c 

(9)

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Hˆ 0  uFˆ y pˆ x   Fˆ z pˆ y  acs  x ˆ x  Vdc ,

 0   0  , uF   F J 0 2a0  , acs x   J 0 2a0  s x  .  c0 

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After use of the next unitary transformation:

1 Hˆ  Uˆ  Hˆ 0Uˆ ,   Uˆ  0 , Uˆ  ˆ x  ˆ y ˆ x  ˆ z  , 2

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we obtain instead of Eq. (9) the next transformed Hamiltonian (10)

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Hˆ  uFˆ x pˆ x   Fˆ y pˆ y  acs  x ˆ z  Vdc  x  .

Thus in non-resonant approximation the effective Hamiltonian (10) of ac-driven GSL corresponds to the Dirac electron with anisotropic Fermi velocity moving through the periodically arranged

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barriers with profile renormalized by the ac-field action and represented by the function acs  x  .

3. Kronig-Penney model Now we consider the functions  s  x  and Vdc  x  written in the form of square-wave barriers as it is shown in Fig. 1. The widths of well and barrier are defined as d w and d b correspondingly,

ACCEPTED MANUSCRIPT so

d  dw  db .

The

equation

Hˆ  r    eff  r 

admits

solutions

in

the

form

 r    nw ,b x expip y y  . Here Hˆ is the effective Hamiltonian (11), functions acs x  and Vdc x  are acs  x   J 0 2a0 1 and Vdc  x   0 , if n  1d  x  d w  n  1d ,

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acs  x   J 0 2a0  2 and Vdc  x   V0 , if d w  n  1d  x  nd . The parts of the wave functions defined as  nw , b  x  correspond to the electron states in the well and barrier with number n and satisfy the boundary conditions

 nw d w  0   nb d w  0 ,  nb d  0   nw1  0 .

Dispersion relation in this case is derived from the equation Re T11  cos p x d  , where T11 is the transfer matrix element located in the first row and first column. Transfer matrix can be written as

ˆ 1 0 ˆ d  ˆ 1 d  ˆ d  , Tˆ   1 2 2 w 1 w

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where

1  iˆ z p 1, 2 x u F p 2  i F p y u p  i F p y  , 1  F 1 , 2  , * e  1, 2   eff  J 0 2a0 1  eff  V0  J 0 2a0  2

1 ˆ  x     1, 2  1, 2 

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2 uF2 p12   eff   F2 p y2  J 02 2a0 21 , uF2 p22   eff  V0    F2 p y2  J 02 2a0 22 .

2





uF2 p12  uF2 p22  V02  J 02 2a0  22  21 sin  p1d w sin  p2 d b   2uF2 p1 p2

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

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Having found the real part of T11 we obtain

 cos p1d w  cos p2 d b   cos p x d  .

(11)

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If ac-field is absent ( a0  0 ) and 1   2  0 formula (11) yields the result [37,38]. Also if a0  0 and 1  0 it gives the dispersion relation [39]. The numerical analysis shows that in the presence of ac-field the widths of the dynamically modified gaps and conduction minibands change with amplitude a0 . The dependences of the dynamically modified gap between valence and conduction minibands  g* and of the width of the first conduction miniband  c* on the amplitude a0 are shown in Figs. 2 and 3 for the next

ACCEPTED MANUSCRIPT parameters values: 21  0.053 eV, 2 2  0.26 eV (such values are typical for substrate h-BN/SiC [41,42]),  F  108 cm/s, d  2·10–6 cm, d b d w  4, V0  0 .

4. Dynamical gap generation in the vicinity of extra Dirac points

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The emergence of so-called extra Dirac points in the electronic structure of the GSL had been predicted in Ref. [38] where the Kronig-Penney model of the potential Vdc  x  had been suggested for the gapless graphene ( 1   2  0 ). The ac-field was absent in the problem investigated in Ref. [38]. Apart from the usual Dirac point ( p x  p y  0 ) the extra Dirac points are arranged along the

p y axis and their location determined by the relations [38]:

 F2 p y2,n 

V02 4 2 F2 n 2  , 4 d2

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where n is nonzero integer, well and barrier widths are assumed to be equal ( d w  d b ). Indeed, with the help of Eq. (11) it easy to check that in the absence of ac-field ( a0  0 ) the conduction and

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valence bands meet at points with px  0 and p y  p y ,n , wherein energy is   V0 2 . Let now ac-field with potential (1) is present ( a0  0 ). If 1   2  0 then Eq. (11) yields

uF2 q12  uF2 q22  V02  q1d   q2 d  qd  q d  sin   sin    cos 1  cos 2   cos p x d  . 2 2uF q1q2  2   2   2   2 

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

(12)

2 Here parameters q1, 2 are determined by the formulas uF2 q12   eff   F2 p y2 , uF2 q22   eff  V0    F2 p y2 .

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2

It is follows from the dispersion relation (12) that electron states in the extra Dirac points (i.e. states

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with px  0 and p y  p y , n ) do not exist. It means the appearance of the gaps  g which are dynamically induced by ac-field in the vicinity of each extra Dirac points said above. In the case of weak ac-field ( a0  1 ) the calculations give the next formula for the gap width

 g ~

32 2 F2 n 2  F p y ,n a02 . V02 d

(13)

ACCEPTED MANUSCRIPT Predicted in this section dynamical appearance of the gap in the vicinity of extra Dirac points of the band structure of GSL due to the ac-field is new result. Dependences of  g on the ac-field amplitude calculated within the Kronig-Penney model are shown in Fig. 4, where dashed line

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corresponds to the approximate formula (13).

5. Tight-binding approximation

In this section we obtain the quasienergy eigenvalues of the equation Hˆ    eff  with use of tight-binding approximation as well as in Ref. [43]. Here operator Hˆ is Hamiltonian (10) written in the case of absence of scalar potential ( V0  0 ). Conditions of tight-binding approximation lead to the analytical expression for the electron quasienergy spectrum

(14)

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 eff   F2 p y2  J 02 2a0  2   2 cos p x d  ,

where parameters  and  are defined by the shape of SL barriers and by the ratio between the barrier width and the well width. For square barriers with d b d w  4 and for the pointed above

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values of 1 ,  2 , and d calculations give  ~ 0.122 eV,  ~ 0.047 eV. If a0  0 then the energy spectrum [43] follows from the formula (14). Thus quasienergy (as well as energy) spectrum is seen

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from Eq. (14) to be nonadditive unlike the conventional semiconductor SL [30]. This fact leads to

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the mutual influence of the dynamics of the electrons moving through the orthogonal directions. The use of the explicit form of the quasienergy spectrum of GSL (14) allows us to find the

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analytical expressions for the field-modified gap  g* and conduction miniband width  c* of ac-driven GSL ( p y  0 ):

 g*  2 J 0 2a0   2   2 ,  c*  J 0 2a0 

   2

2



 2   2 .

(15)

Plotted by the formulas (15) dependences of the parameters  g* and  c* on the amplitude a0 are represented in Figs. 2 and 3 by dashed lines. A good agreement between the numerical analysis performed with the help of dispersion relation (11) and analytical model (14) derived with the use

ACCEPTED MANUSCRIPT of tight-binding approximation is seen from these figures. To decrease the numerical difference between considered models it is necessary to increase the height of the barriers  2 . It can be reached by the use of substrate material which induces in the graphene band structure the gap lager than that of graphene on the SiC-substrate.

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The field-modified gap and conduction miniband width are seen from the Figs. 2 and 3 to change with amplitude a0 . Note that dynamical narrowing of the conduction miniband of conventional semiconductor SL with additive spectrum induced by the high-frequency electric field had been studied in Ref. [18], where electric field vector had been assumed to oscillate parallel to the SL axis. But above we show that nonadditivity of electron spectrum of GSL leads to the dynamical modification of its band structure even if vector of ac electric field is applied perpendicularly to the GSL axis. Observation of such feature of the quasienergy spectrum is

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possible in the real fields [27,28], and the problem of the field-induced modification of the electron spectrum of graphene materials is of interest in connection with the possibilities of band parameters

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controlling.

6. Modification of the negative differential conductivity

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The study of the negative differential conductivity (NDC) is the traditional problem of the investigations of electron transport in structures with SL [44,45]. The interest of researchers to such

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investigations is explained by the possibility of creation of generator of Bloch THz generator. One of the causes of NDC is the Bragg reflections at the SL miniband boundaries [44]. The appearance

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of this effect can be easy explained by next reasoning. Let SL is assumed to be subjected by dc electric field with vector Edc applied through the SL axis Ox . If electrons of SL acquire in dc-field the energy enough to reach the top of the conduction miniband then the intensity Edc satisfies the condition j x Edc ~  c n0  . Here j x is the current density induced by the dc-field,  c is the miniband width,  is the average time between electron collisions, n0 is the charge carriers concentration. It

ACCEPTED MANUSCRIPT is seen from the last condition that the increasing of the dc-field intensity Edc leads to the decreasing of the current density j x . Note that this conclusion is quite correct while the oneminiband model works and it is true that eEdc d   1 .

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In the case when SL (or GSL) is subjected additionally to the ac electric field we have to make the substitution  c   c* for the condition of NDC appearance. Here  c* is the field-modified miniband width. To find the current density corresponding to NDC we use the model (14). After using the second formula (15) for parameter  c* we write

jx ~

n0 2 J 0 2a0  .   Edc

When last formula was being derived we took into account that  2   2 . Thus the slope of the current-voltage characteristic depends on the ac-field amplitude. Similar effect was obtained in Ref.

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[30] for the semiconductors SL with additive spectrum in the case when high-frequency electric field vector oscillated parallel to the SL axis. But here we have shown that ac electric field effects

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on the NDC of GSL even if its intensity vector oscillates perpendicularly to the GSL axis unlike SL

7. Discussion

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with additive spectrum.

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Above the quasienergy spectrum of electrons in GSL subjected to the ac electric field has been calculated. The ac-field is assumed to have nonresonant character. The calculations have been

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performed both within the Kronig-Penney model and within the tight-binding approximation. In the case when V0  0 the explicit form of the quasienergy spectrum have been derived (formula (14)). Nonadditivity of GSL electron spectrum has been shown to lead to the dynamical modification of the band gap and the miniband width even if the vector of ac electric field is applied perpendicularly to the GSL axis. Such feature makes GSL different from the conventional SL with additive spectrum where the effect of ac-field with such polarization leads only to a shift of

ACCEPTED MANUSCRIPT spectrum as a whole. For instance, unlike SL with additive spectrum dynamical effect of EM radiation polarized perpendicularly to the GSL axis leads to the dependence of the slope of the GSL current-voltage characteristic on the ac-field amplitude. This is the advantage of GSL over the conventional SL [18,30,31,44,45] where the current-voltage characteristic modification caused by

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the ac pump field is sensitive to the field polarization and, in particular, it disappears when the pump field is polarized perpendicularly to the SL axis.

Dependences of field-modified gap and miniband width in GSL bad structures on the amplitude a0 calculated within the Kronig-Penney model and with the help of formulas (15) derived in the tight-binding approximation are in quite good agreement. Difference between the results obtained within such models can be decreased by increasing of the height  2 of the SL barriers.

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Besides, in the vicinity of extra Dirac points of GSL band structure the gap  g has been shown above to be dynamically induced by the ac-field. For weak ac-fields this gap increases as a02 .

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It is quite seen from the Fig. 4 (the region of dashed line). For arbitrary values of amplitude a0

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except the vicinity of the zeros of Bessel function ( J 0 2a0   0 ) the gap  g oscillates with a0 .

Acknowledgements

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The work was supported with the funding of the Ministry of Education and Science of the

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Russian Federation within the base part of the State task № 2014/411 (Project Code: 3154).

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ACCEPTED MANUSCRIPT Figure captions

Fig. 1. Spatial periodic modulation of the band gap in graphene

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Fig. 2. Dependences of the field-modified gap between valence and conduction minibands  g* on the ac-field amplitude. Solid line corresponds to the Kronig-Penney model, dashed line corresponds to the tight-binding approximation

Fig. 3. Dependences of the field-modified width of the first conduction miniband  c* on the ac-field amplitude. Solid line corresponds to the Kronig-Penney model, dashed line corresponds to the tight-

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binding approximation

Fig. 4. Dependences of the gap induced dynamically by the ac-field in the vicinity of the extra

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approximate formula (13)

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Dirac points with n   1 on the amplitude. V0  0.45 eV, dashed line corresponds to the

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