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Electronic band gaps and transport in Cantor graphene superlattices
Accepted Manuscript Electronic band gaps and transport in Cantor graphene superlattices Yi Xu, Ying He, Yanfang Yang, Huifang Zhang PII: DOI: Reference:
S0749-6036(15)00002-6 http://dx.doi.org/10.1016/j.spmi.2014.12.031 YSPMI 3548
To appear in:
Superlattices and Microstructures
Received Date: Revised Date: Accepted Date:
4 November 2014 29 December 2014 31 December 2014
Please cite this article as: Y. Xu, Y. He, Y. Yang, H. Zhang, Electronic band gaps and transport in Cantor graphene superlattices, Superlattices and Microstructures (2015), doi: http://dx.doi.org/10.1016/j.spmi.2014.12.031
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Electronic band gaps and transport in Cantor graphene superlattices Yi Xu1,2, Ying He1,a , Yanfang Yang1 and Huifang Zhang1 1 Department of Physics, College of Sciences, Shanghai University, Shanghai 200444, China 2 Engineering Product Development, Singapore University of Technology and Design, 20 Dover Drive, Singapore 138682, Singapore
Abstract. The electronic band gap and transport in Cantor graphene superlattices are investigated theoretically. It is found that such fractal structure can possess an unusual Dirac point located at the energy corresponding to the zero-averaged wave number ( zero k ). The location of the Dirac point shifts to lower energy with the increase of order number. The zero k gap is robust against the lattice constants and less sensitive to the incidence angle. Moreover, multi-Dirac-points may appear by adjusting the lattice constants and the order, and an expression for their location is derived. The control of electron transport in such fractal structure may lead to potential applications in graphene-based electronic devices.
Keywords: graphene superlattices; Cantor set; band gap; electron transport
a
e-mail: [email protected]
1 Introduction The electronic band gap and transport in graphene have attracted much attention due to its novel electronic properties since the extremely thin carbon material was experimentally realized by Novoselov et al. in 2004 [1]. In graphene, the quasiparticles close to the Dirac point (DP) (often referred to as K and K ) of the Brillouin zone posses a gapless linear energy spectrum. As a consequence, graphene exhibits numerous unique electronic and transport properties, such as half-integer quantum Hall effect [2-4], the minimal conductance [2], the zitterbewegung [5, 6], Klein tunneling [7] and atomic collapse [8, 9]. Since superlattices are very successful in controlling the electronic transport, more and more theoretical [10-30] and experimental [31-33] researches focus on the graphene superlattices (GSLs) with electrostatic potential [10-25] and magnetic barriers [26-30]. Recently, several researches about the zero-averaged wave number ( zero k ) gap associated with the new DP in GSLs have been reported [11-23, 26]. The zero k gap in GSLs is similar to the zero-averaged refractive index bandgap in photonic crystals containing left-handed metamaterials [34]. Contrary to Bragg gaps, the zero k gap associated with the new DP is robust against the lattice constant, structural disorder [11], and external magnetic field [26], which can be applicable to control the electronic transport in GSLs. The electronic band gaps and transport in quasi-periodic GSLs, such as Fibonacci [17], Thue-Morse [18-20], and double-periodic [21] GSLs, have also been investigated theoretically. It is found that the zero k gap also exists in these quasi-periodic GSLs, it is less sensitive to the incidence angle, and robust against the lattice constants. The electronic band gaps and transport properties in these quasi-periodic GSLs may facilitate the development of graphene-based electronic devices. Cantor set, a fractal structure [35], is a homogeneous and self-similar geometric object, which can be used to describe many physical phenomena. As the simplest fractal structure, the triadic Cantor set has been widely studied on the transmission properties of light and electrons [36-38], quantum scattering and tunneling [39, 40]. The Cantor set is different from
the periodic and quasi-periodic sequences, such as Fibonacci, Thue-Morse, and double-periodic sequence. It is also interesting to study the electronic band gaps and transport in Cantor GSLs. Sun et al [41] have studied the transmission of electrons tunneling through GSLs with fractal and periodic magnetic barriers. Most recently, Liu et al [42] have investigated the spin-polarized transport properties of electrons tunneling through a fractal graphene superlattice sandwiched between two ferromagnetic graphene electrodes, and the spin conductance and the tunneling magnetoresistance of the Ferromagnetic Graphene/Fractal Graphene Superlattice/Ferromagnetic Graphene junction have been compared with those of the Ferromagnetic Graphene/Symmetric Periodic Graphene Superlattice/Ferromagnetic Graphene junction. Motivated by these unusual electronic properties in the periodic and quasi-periodic GSLs, we investigate the electronic band gaps and transport in the graphene-based Cantor sequence. In this paper, we find that the zero k gap and DP also exist in the graphene-based Cantor sequence. Different from the Fibonacci case, the Cantor sequence do not visualize the Cantor-like distribution of energy bands. And the location of DP shifts to lower energy region as the order n increases. We also derived an expression for the location of multi-Dirac-points.
2 Theoretical model A triadic Cantor sequence can be out-bound with the single layer width is given, and the n th triadic Cantor sequence is defined by the rule: Sn Sn1Bn Sn1 for n 2 , while
S0 A , S1 ABA , as shown in figure 1(a). Bn for n th layer denotes the layer B with a width of dBn 3n1 dB . Here, the element A (B) denotes a barrier VA (well VB ) with the width dA (dB ) . The triadic Cantor sequence also can be in-bound with the total width of the structure is given at the beginning, as shown in figure 1(b). First take a seed to be a bulk of layer A with a given width (in figure 1(b), the initial width is the width of 2-th Cantor structure in out-bound form). Then divide the seed into three equal parts, and replace the
center part of the seed with layer B. Then repeat the same procedure over all remaining initial layer, as if they were seeds. And an n th in-bound form Cantor structure is obtained when these steps recurred n times. It is notable that the ratio of width of layer A and B, dA dB , is always equal to 1 in in-bound form, while it can be adjusted in out-bound form. N A 2n and N B 3n 2n are the numbers of barrier A and well B for the n th Cantor sequence both in in-bound and out-bound form. In this work, we will focus on the out-bound form. The Dirac-like electrons moving inside a monolayer graphene in the presence of the electrostatic potential V ( x) can be described by the Dirac-like equation:
i
vF V ( x) ( x, y) E( x, y) ,
(1)
where ( x , y ) are the Pauli matrices, vF 106 m s is the Fermi velocity, and ( A , B )T is a two-component pseudospinor wave function. Due to the translation
invariance in the
y ik y y
A,B ( x, y) A,B ( x)e
direction, the wave function A,B
can be presented as
( x) . The smooth enveloping function A at x and x x in B ( x )
the jth potential can be connected with a transfer matrix [11, 25]: cos(q j x j ) cos j M j (x, E , k y ) sin(q j x) i cos j
Here, the longitudinal wave vector q j i k y2 k 2j . k j ( E V j )
sin(q j x)
cos j . cos(q j x j ) cos j i
q j sign(k j ) k 2j k y2
for
(2)
k 2j k y2 , otherwise
vF is the wave vector inside the potential V j , and
j arcsin(k y k j ) is the angle in the jth potential. Then the transmission coefficient is given by:
t ( E, k y )
( x22e
i0
2cos0
x11eie ) x12ei(e 0 ) x21
,
(3)
where 0 (e ) is the incident (exit) angle of the electron wave, and xij (i, j 1,2) is the matrix element of
X Sn Nj 1 M j (d j , E, k y ) , which is the entire transfer matrix
connecting the incident and exit wave functions, N NA NB 3n is the total number of 2
layers of the Cantor GSL. Then, the transmission probability is T t , and the total conductance G of the system at zero temperature can be obtained according to the Landauer–Büttiker formula [43, 44], G =G0
2
- 2
T cos0d0 , where G0 2e2 ELy ( ) is
taken as the conductance unit with L y the sample size along the y direction. The Fano factor is given by [45]: F =
2
- 2
T (1 T )cos0d0
2
- 2 T cos0d0 .
3 Results and discussion Treating an
n th
Cantor sequence as a unit cell, then we can derive
cos( x n ) Tr X Sn 2 from Bloch’s theorem, where x is the x component of
Bloch wave vector and n N Ad A N B d B . Figures 2(a) and (b) present the trace maps for Cantor GSLs with the change of order n at k y 0.01nm1 , where (a) d A d B 20nm , (b) d A d B 25nm . It is found that there are several broad forbidden gaps opening for each generation, and the passing bands are split into more and more subbands as the order n increases. Different from the results for Fibonacci sequence [17], the distribution of energy bands does not exhibit Cantor-like set although the superlattice structure is a Cantor-set. Among these forbidden gaps, the centre positions of the zero k gaps, which are denoted by a solid red line, are almost the same for the same n th Cantor sequence in figures 2(a) and (b). These zero k gaps are different from Bragg gaps since the positions of these gaps are shifted with the change of the lattice constants, as shown in figures 2(a) and (b). Corresponding to figures 2(a) and (b), the electronic band structure of 2-th Cantor GSL is presented in figures 2(d) and (e), respectively. It can be seen from the figures that a new DP appears inside the zero k gap. However, unlike the graphene-based Thue-Morse [18-20] and double-periodic [21] sequence, the position of zero k gap in Cantor GSLs for each generation is different, and shifts to lower energy region with the increase of order, as shown in figure 2.
To understand the zero k gap better, equation (4) gives the electronic dispersion of the second Cantor sequence (S2 = ABABBBABA), at any incidence angle cos( x 2 ) Tr X S2 2 cos(4qA d A )cos(5qB d B )
. sin A sin B 1 sin(4qA d A )sin(5qB d B ) cos A cos B
(4)
Here, 2 4d A 5d B . At the normal incidence, i.e. 0 =0 , k y =0 , the usual DP is given by
4qA dA 5qBdB . That is
E
4VA d A 5VB d B . 4d A 5d B
(5)
Actually, the location of the new DP can be derived from the electronic dispersion of the
n th Cantor sequence cos( x n ) Tr X Sn 2 cos( N A qA d A )cos( N B qB d B )
, sin A sin B 1 sin( N A qA d A )sin( N B qB d B ) cos A cos B
(6)
where n NA d A NBd B . At the normal incidence, the usual DP is given by
NA qA dA NBqBdB .
(7)
Equation (7) is equivalent to N
k k jd j j 1
N
d j 1
j
0,
(8)
which is valid for periodic and aperiodic GSLs [11, 16-21]. In the Cantor GSLs, the new DP locates at
En
N AV A N VB Bd Bd N A N Bd B d A
2n VA ( 3n 2nV )B d B d 2n ( 3n 2n d)B d A
A
.
(9)
A
Equations (5) and (9) are the same when n 2 . Equation (9) also indicates that the position of new DP depends on the order n and the ratio d B d A . As shown in figures 2(a), (b) and (c), the positions of zero k gaps for each generation are different, and shift to lower energy region with the increase of order n . The position of DP is E 22.22meV since
d A d B for the 2-th Cantor GSL, while it moves to E 14.29meV for d B d A 2 , as
shown in figure 2(f). Furthermore, the slope of the band edges near the DP turns smaller with the increase of d A (d B ) , as is seen in figures 2(d), (e) and (f). Equation (9) has demonstrated that the zero k gap depends on the ratio of lattice parameters, d B d A , rather than the lattice constants themselves. It also can be seen from figure 3(a), which shows the effect of the lattice parameters on the transmission spectra for the 2-th Cantor GSL, (ABABBBABA) 8. The position of the zero k gap changes as
d B d A varies. From figure 3(b), it is obvious that the zero k gap is weakly dependent on the incidence angle, while the Bragg gap is sensitive to the incidence angle. The dependence of zero k gap on the lattice constants and incidence angle is valid for periodic and aperiodic GSLs [11, 17, 18, 20, 21]. In fact, the Cantor GSLs is a periodic structure with defects in it. For instance, the 2-th Cantor GSL, (ABABBBABA)8 can be treated as that three impurity layers B are inserted into a unit cell of a periodic structure (ABAABAABA) 8. Namely, this fractal structure can be treated as a disordered periodic structure. Thus, the results in figure 3 indicate that the zero k gap is robust against the structural disorder, which has been demonstrated in Ref. 11. In addition, many types of long wavelength disorder, such as ripples, wrinkles and dislocations, influence the electrons as effective gauge fields [46-48], which modify the electronic properties. As discussed in Ref. [26], the zero k gap associated with the new DP is robust against external magnetic field, so we do not discuss the effect of the gauge fields induced by the elastic deformation of graphene strip. Although a scalar potential is also created by the elastic deformation of the graphene strip, it is relatively small [46], which can be neglected. It has been demonstrated that the multi-DPs could appear in the GSLs with periodic and aperiodic structures [11, 16, 18-21]. Figure 4(a) demonstrates that the extra DPs located at k y 0 can also emerge in the Cantor GSLs, where k y 0.01nm1 , and dA dB . As shown
in figure 4(a), the zero k gap associated with the multi-DPs, oscillates with changing the lattice constants, while the other gaps are significantly shifted.
The location of extra DP for the n th Cantor GSLs can be derived from Equation (6). When
0 0 , that is, at oblique incidence, the extra DPs can appear if
NA qA dA NBqBdB =m is satisfied, where m is a positive integer. Then the location of the extra DP is
E
VA +VB ( vF m )2 1 1 , 2 2(VA VB ) N A d A 2 N B d B 2
(10)
and 2
k y ,m
E VA m vF N A d A 2
E VB m vF N B d B
2
.
(11)
2
When d A d B d , it can be seen from equation (8) that E
VA +VB , which is different 2
from the periodic [11] and Thue-Morse GSLs [18-20]. When dA dB 40nm , a pair of new DPs emerges at k y 0 in the 2-th Cantor GSLs, and the energy of new DPs is slightly different from the DP at k y =0 , as presented in figure 4(b). Equation (11) also shows that the location of extra DP depends on the order n since N A 2n and N B 3n 2n depend on
n. To demonstrate the number and locations of the DPs, we show a transmission spectrum for the 2-th Cantor GSL in figure 5 where the energy is fixed at E 22.22meV . It can be seen from figure 5 that some transmission peaks appear at some certain angles, which correspond to the DP. When d A d B 20nm(25nm) , a transmission peak located at the incidence angle 0 =0 , which corresponds to the figure 2(d) (2(f)). When the lattice constants are d A d B 40nm , a pair of tunneling states with unity transmission probability appear in the transmission spectrum, which means extra DPs appear in the band structure, as shown in figure 4(b). Furthermore, the difference of electronic band gaps between the out-bound form (figure
1(a)) and in-bound form (figure 1(b)) is presented in figure 6. We take d A, B 20nm in figures 6(a) and (c), while d A, B 20 3nm in figure 6(b), d A, B 20 9nm in figure 6(d). It can be seen from the figure that the location of new DP is independent of the lattice constants with the same order n , and the electronic band structure of out-bound form is more complex than that of in-bound form. It is notable that the extra DPs located at k y 0 can also emerge in the 3-th Cantor GSLs in out-bound form, as shown in figure 6(a), where the locations of additional DPs are different from the DP in the case of normal incidence. It indicates that the multi-DP can be controlled by adjusting the order n (see equation (10) or (11)), which provides the flexibility to control the electron transport in Cantor GSLs by adjusting the order. The conductance and the Fano factor in the 2-th Cantor GSL, (ABABBBAAA)8, is presented in figure 7. It shows that the angular-averaged conductance curve reaches its minimum at the DP, and forms a linear cone around the DP. Meanwhile, the Fano factor reaches the value of 1/3 approximately at the corresponding DP’s location or conductance minimum. Moreover, from figure 7, we can see that the conducatance and Fano factor shift with the ratio d B d A , which indicates that the conductance and Fano factor of Cantor GSLs can be modulated by adjusting the ratio of lattice constants. In addition, we should point out that the trace map and corresponding electronic transport properties in Cantor GSLs are different from the Fibonacci, Thue-Morse, and double-periodic case. The ratio N A N B in Thue-Morse sequence is always equal to unit, while it depends on the variable order n for Fibonacci and double-periodic case, as well as the Cantor sequence. However, the position of new DP for Fibonacci GSLs will become stable for
n 1 since limn N A N B 1 5
2 while limn N A N B 2 for double-periodic
sequence. It is interesting that limn N A N B 0 for the Cantor case, thus the location of new DP in Cantor sequence tends to zero for n 1 . 4 Summary In summary, we have investigated the electronic band gaps and transport in the
graphene-based Cantor superlattices with the transfer matrix method. It is found that the
zero k gap associated with the DP is robust against the lattice constants and less sensitive to the incidence angle, which can be applicable to control the electron transport in Cantor GSLs. The position of zero k gap in Cantor GSLs shifts to lower energy region with the increase of order n , and tends to zero for n 1 , which is different from the periodic, Fibonacci, Thue-Morse and double-periodic GSLs. Furthermore, multi-DPs can be controlled by adjusting the lattice constants and the order n , and an expression for their location is derived. We hope the results obtained in such fractal structure will have applications in various graphene-based electron devices, such as graphene-based electronic omnidirectional reflector and filters. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant Nos. 11204170, 61108010).
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Figures
Fig. 1. Example of Cantor GSLs, (a) out-bound form, (b) in-bound form. The black and white blocks represent the barrier A and well B, respectively. (c) The schematic profiles of the potentials VA and VB corresponding to 2-th Cantor structure both in (a) and (b).
Fig. 2. Trace-maps for Cantor GSL as a function of order n under (a) dA dB 20nm , (b)
dA dB 25nm ,
(c)
2dA dB 40nm
with VA 50meV ,
VB 0meV
and
k y 0.01nm1 . (d), (e) and (f) are band structures for n 2 corresponding to (a), (b) and (c), respectively. The horizontal solid red line represents the location of new Dirac point, and the white areas are band gaps while the solid areas are passing bands.
Fig. 3. Transmission spectra for Cantor GSL, (ABABBBABA)8 , with (a) different widths of barriers and wells at 0 10 . Here, solid black line for dA dB 15nm , dotted blue line for dA dB 20nm , and dashed-dotted red line for 2dA dB 40nm ; (b) different incidence angles, 0 10
(solid black line), 15
(blue dotted blue line) and 20
(dashed-dotted red line) with the lattice constant dA dB 20nm . Other parameters are the same as those in figure 2.
Fig. 4. (a) Band gaps depending on the lattice constant d with k y 0.01nm1 , here
dA dB d . (b) Electronic band structures for dA dB 40nm . The order is n 2 , other parameters are the same as those in figure 2.
Fig. 5. Transmission spectrum for a S2 = ABABBBABA Cantor GSL with different incidence angle at E 22.22meV . Here, dA dB d , and the other parameters are the same as those in figure 2.
Fig. 6. Electronic band structures for out-bound form (a) and (c), and in-bound form (b) and (d). The order n 3 for (a), (b), while n 4 for (c), (d). The lattice constant in (a), (c) is
dA dB 20nm , while dA dB 20 3nm for (b), dA dB 20 9nm for (d). Other parameters are the same as those in figure 2.
Fig. 7. Conductance (a) and (b) and Fano factor (c) and (d) versus Fermi energy in Cantor GSL, (ABABBBAAA)8 . Other parameters are the same as those in Fig. 2.
Highlights The Cantor graphene superlattices are investigated theoretically. Cantor fractal structure can possess an unusual Dirac point. The Dirac corresponds to the zero-averaged wave number. Multi-Dirac-points appear by adjusting the lattice constants and the order. The electron transport in Cantor fractal structure can be controlled.
S0749-6036(15)00002-6 http://dx.doi.org/10.1016/j.spmi.2014.12.031 YSPMI 3548
To appear in:
Superlattices and Microstructures
Received Date: Revised Date: Accepted Date:
4 November 2014 29 December 2014 31 December 2014
Please cite this article as: Y. Xu, Y. He, Y. Yang, H. Zhang, Electronic band gaps and transport in Cantor graphene superlattices, Superlattices and Microstructures (2015), doi: http://dx.doi.org/10.1016/j.spmi.2014.12.031
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Electronic band gaps and transport in Cantor graphene superlattices Yi Xu1,2, Ying He1,a , Yanfang Yang1 and Huifang Zhang1 1 Department of Physics, College of Sciences, Shanghai University, Shanghai 200444, China 2 Engineering Product Development, Singapore University of Technology and Design, 20 Dover Drive, Singapore 138682, Singapore
Abstract. The electronic band gap and transport in Cantor graphene superlattices are investigated theoretically. It is found that such fractal structure can possess an unusual Dirac point located at the energy corresponding to the zero-averaged wave number ( zero k ). The location of the Dirac point shifts to lower energy with the increase of order number. The zero k gap is robust against the lattice constants and less sensitive to the incidence angle. Moreover, multi-Dirac-points may appear by adjusting the lattice constants and the order, and an expression for their location is derived. The control of electron transport in such fractal structure may lead to potential applications in graphene-based electronic devices.
Keywords: graphene superlattices; Cantor set; band gap; electron transport
a
e-mail: [email protected]
1 Introduction The electronic band gap and transport in graphene have attracted much attention due to its novel electronic properties since the extremely thin carbon material was experimentally realized by Novoselov et al. in 2004 [1]. In graphene, the quasiparticles close to the Dirac point (DP) (often referred to as K and K ) of the Brillouin zone posses a gapless linear energy spectrum. As a consequence, graphene exhibits numerous unique electronic and transport properties, such as half-integer quantum Hall effect [2-4], the minimal conductance [2], the zitterbewegung [5, 6], Klein tunneling [7] and atomic collapse [8, 9]. Since superlattices are very successful in controlling the electronic transport, more and more theoretical [10-30] and experimental [31-33] researches focus on the graphene superlattices (GSLs) with electrostatic potential [10-25] and magnetic barriers [26-30]. Recently, several researches about the zero-averaged wave number ( zero k ) gap associated with the new DP in GSLs have been reported [11-23, 26]. The zero k gap in GSLs is similar to the zero-averaged refractive index bandgap in photonic crystals containing left-handed metamaterials [34]. Contrary to Bragg gaps, the zero k gap associated with the new DP is robust against the lattice constant, structural disorder [11], and external magnetic field [26], which can be applicable to control the electronic transport in GSLs. The electronic band gaps and transport in quasi-periodic GSLs, such as Fibonacci [17], Thue-Morse [18-20], and double-periodic [21] GSLs, have also been investigated theoretically. It is found that the zero k gap also exists in these quasi-periodic GSLs, it is less sensitive to the incidence angle, and robust against the lattice constants. The electronic band gaps and transport properties in these quasi-periodic GSLs may facilitate the development of graphene-based electronic devices. Cantor set, a fractal structure [35], is a homogeneous and self-similar geometric object, which can be used to describe many physical phenomena. As the simplest fractal structure, the triadic Cantor set has been widely studied on the transmission properties of light and electrons [36-38], quantum scattering and tunneling [39, 40]. The Cantor set is different from
the periodic and quasi-periodic sequences, such as Fibonacci, Thue-Morse, and double-periodic sequence. It is also interesting to study the electronic band gaps and transport in Cantor GSLs. Sun et al [41] have studied the transmission of electrons tunneling through GSLs with fractal and periodic magnetic barriers. Most recently, Liu et al [42] have investigated the spin-polarized transport properties of electrons tunneling through a fractal graphene superlattice sandwiched between two ferromagnetic graphene electrodes, and the spin conductance and the tunneling magnetoresistance of the Ferromagnetic Graphene/Fractal Graphene Superlattice/Ferromagnetic Graphene junction have been compared with those of the Ferromagnetic Graphene/Symmetric Periodic Graphene Superlattice/Ferromagnetic Graphene junction. Motivated by these unusual electronic properties in the periodic and quasi-periodic GSLs, we investigate the electronic band gaps and transport in the graphene-based Cantor sequence. In this paper, we find that the zero k gap and DP also exist in the graphene-based Cantor sequence. Different from the Fibonacci case, the Cantor sequence do not visualize the Cantor-like distribution of energy bands. And the location of DP shifts to lower energy region as the order n increases. We also derived an expression for the location of multi-Dirac-points.
2 Theoretical model A triadic Cantor sequence can be out-bound with the single layer width is given, and the n th triadic Cantor sequence is defined by the rule: Sn Sn1Bn Sn1 for n 2 , while
S0 A , S1 ABA , as shown in figure 1(a). Bn for n th layer denotes the layer B with a width of dBn 3n1 dB . Here, the element A (B) denotes a barrier VA (well VB ) with the width dA (dB ) . The triadic Cantor sequence also can be in-bound with the total width of the structure is given at the beginning, as shown in figure 1(b). First take a seed to be a bulk of layer A with a given width (in figure 1(b), the initial width is the width of 2-th Cantor structure in out-bound form). Then divide the seed into three equal parts, and replace the
center part of the seed with layer B. Then repeat the same procedure over all remaining initial layer, as if they were seeds. And an n th in-bound form Cantor structure is obtained when these steps recurred n times. It is notable that the ratio of width of layer A and B, dA dB , is always equal to 1 in in-bound form, while it can be adjusted in out-bound form. N A 2n and N B 3n 2n are the numbers of barrier A and well B for the n th Cantor sequence both in in-bound and out-bound form. In this work, we will focus on the out-bound form. The Dirac-like electrons moving inside a monolayer graphene in the presence of the electrostatic potential V ( x) can be described by the Dirac-like equation:
i
vF V ( x) ( x, y) E( x, y) ,
(1)
where ( x , y ) are the Pauli matrices, vF 106 m s is the Fermi velocity, and ( A , B )T is a two-component pseudospinor wave function. Due to the translation
invariance in the
y ik y y
A,B ( x, y) A,B ( x)e
direction, the wave function A,B
can be presented as
( x) . The smooth enveloping function A at x and x x in B ( x )
the jth potential can be connected with a transfer matrix [11, 25]: cos(q j x j ) cos j M j (x, E , k y ) sin(q j x) i cos j
Here, the longitudinal wave vector q j i k y2 k 2j . k j ( E V j )
sin(q j x)
cos j . cos(q j x j ) cos j i
q j sign(k j ) k 2j k y2
for
(2)
k 2j k y2 , otherwise
vF is the wave vector inside the potential V j , and
j arcsin(k y k j ) is the angle in the jth potential. Then the transmission coefficient is given by:
t ( E, k y )
( x22e
i0
2cos0
x11eie ) x12ei(e 0 ) x21
,
(3)
where 0 (e ) is the incident (exit) angle of the electron wave, and xij (i, j 1,2) is the matrix element of
X Sn Nj 1 M j (d j , E, k y ) , which is the entire transfer matrix
connecting the incident and exit wave functions, N NA NB 3n is the total number of 2
layers of the Cantor GSL. Then, the transmission probability is T t , and the total conductance G of the system at zero temperature can be obtained according to the Landauer–Büttiker formula [43, 44], G =G0
2
- 2
T cos0d0 , where G0 2e2 ELy ( ) is
taken as the conductance unit with L y the sample size along the y direction. The Fano factor is given by [45]: F =
2
- 2
T (1 T )cos0d0
2
- 2 T cos0d0 .
3 Results and discussion Treating an
n th
Cantor sequence as a unit cell, then we can derive
cos( x n ) Tr X Sn 2 from Bloch’s theorem, where x is the x component of
Bloch wave vector and n N Ad A N B d B . Figures 2(a) and (b) present the trace maps for Cantor GSLs with the change of order n at k y 0.01nm1 , where (a) d A d B 20nm , (b) d A d B 25nm . It is found that there are several broad forbidden gaps opening for each generation, and the passing bands are split into more and more subbands as the order n increases. Different from the results for Fibonacci sequence [17], the distribution of energy bands does not exhibit Cantor-like set although the superlattice structure is a Cantor-set. Among these forbidden gaps, the centre positions of the zero k gaps, which are denoted by a solid red line, are almost the same for the same n th Cantor sequence in figures 2(a) and (b). These zero k gaps are different from Bragg gaps since the positions of these gaps are shifted with the change of the lattice constants, as shown in figures 2(a) and (b). Corresponding to figures 2(a) and (b), the electronic band structure of 2-th Cantor GSL is presented in figures 2(d) and (e), respectively. It can be seen from the figures that a new DP appears inside the zero k gap. However, unlike the graphene-based Thue-Morse [18-20] and double-periodic [21] sequence, the position of zero k gap in Cantor GSLs for each generation is different, and shifts to lower energy region with the increase of order, as shown in figure 2.
To understand the zero k gap better, equation (4) gives the electronic dispersion of the second Cantor sequence (S2 = ABABBBABA), at any incidence angle cos( x 2 ) Tr X S2 2 cos(4qA d A )cos(5qB d B )
. sin A sin B 1 sin(4qA d A )sin(5qB d B ) cos A cos B
(4)
Here, 2 4d A 5d B . At the normal incidence, i.e. 0 =0 , k y =0 , the usual DP is given by
4qA dA 5qBdB . That is
E
4VA d A 5VB d B . 4d A 5d B
(5)
Actually, the location of the new DP can be derived from the electronic dispersion of the
n th Cantor sequence cos( x n ) Tr X Sn 2 cos( N A qA d A )cos( N B qB d B )
, sin A sin B 1 sin( N A qA d A )sin( N B qB d B ) cos A cos B
(6)
where n NA d A NBd B . At the normal incidence, the usual DP is given by
NA qA dA NBqBdB .
(7)
Equation (7) is equivalent to N
k k jd j j 1
N
d j 1
j
0,
(8)
which is valid for periodic and aperiodic GSLs [11, 16-21]. In the Cantor GSLs, the new DP locates at
En
N AV A N VB Bd Bd N A N Bd B d A
2n VA ( 3n 2nV )B d B d 2n ( 3n 2n d)B d A
A
.
(9)
A
Equations (5) and (9) are the same when n 2 . Equation (9) also indicates that the position of new DP depends on the order n and the ratio d B d A . As shown in figures 2(a), (b) and (c), the positions of zero k gaps for each generation are different, and shift to lower energy region with the increase of order n . The position of DP is E 22.22meV since
d A d B for the 2-th Cantor GSL, while it moves to E 14.29meV for d B d A 2 , as
shown in figure 2(f). Furthermore, the slope of the band edges near the DP turns smaller with the increase of d A (d B ) , as is seen in figures 2(d), (e) and (f). Equation (9) has demonstrated that the zero k gap depends on the ratio of lattice parameters, d B d A , rather than the lattice constants themselves. It also can be seen from figure 3(a), which shows the effect of the lattice parameters on the transmission spectra for the 2-th Cantor GSL, (ABABBBABA) 8. The position of the zero k gap changes as
d B d A varies. From figure 3(b), it is obvious that the zero k gap is weakly dependent on the incidence angle, while the Bragg gap is sensitive to the incidence angle. The dependence of zero k gap on the lattice constants and incidence angle is valid for periodic and aperiodic GSLs [11, 17, 18, 20, 21]. In fact, the Cantor GSLs is a periodic structure with defects in it. For instance, the 2-th Cantor GSL, (ABABBBABA)8 can be treated as that three impurity layers B are inserted into a unit cell of a periodic structure (ABAABAABA) 8. Namely, this fractal structure can be treated as a disordered periodic structure. Thus, the results in figure 3 indicate that the zero k gap is robust against the structural disorder, which has been demonstrated in Ref. 11. In addition, many types of long wavelength disorder, such as ripples, wrinkles and dislocations, influence the electrons as effective gauge fields [46-48], which modify the electronic properties. As discussed in Ref. [26], the zero k gap associated with the new DP is robust against external magnetic field, so we do not discuss the effect of the gauge fields induced by the elastic deformation of graphene strip. Although a scalar potential is also created by the elastic deformation of the graphene strip, it is relatively small [46], which can be neglected. It has been demonstrated that the multi-DPs could appear in the GSLs with periodic and aperiodic structures [11, 16, 18-21]. Figure 4(a) demonstrates that the extra DPs located at k y 0 can also emerge in the Cantor GSLs, where k y 0.01nm1 , and dA dB . As shown
in figure 4(a), the zero k gap associated with the multi-DPs, oscillates with changing the lattice constants, while the other gaps are significantly shifted.
The location of extra DP for the n th Cantor GSLs can be derived from Equation (6). When
0 0 , that is, at oblique incidence, the extra DPs can appear if
NA qA dA NBqBdB =m is satisfied, where m is a positive integer. Then the location of the extra DP is
E
VA +VB ( vF m )2 1 1 , 2 2(VA VB ) N A d A 2 N B d B 2
(10)
and 2
k y ,m
E VA m vF N A d A 2
E VB m vF N B d B
2
.
(11)
2
When d A d B d , it can be seen from equation (8) that E
VA +VB , which is different 2
from the periodic [11] and Thue-Morse GSLs [18-20]. When dA dB 40nm , a pair of new DPs emerges at k y 0 in the 2-th Cantor GSLs, and the energy of new DPs is slightly different from the DP at k y =0 , as presented in figure 4(b). Equation (11) also shows that the location of extra DP depends on the order n since N A 2n and N B 3n 2n depend on
n. To demonstrate the number and locations of the DPs, we show a transmission spectrum for the 2-th Cantor GSL in figure 5 where the energy is fixed at E 22.22meV . It can be seen from figure 5 that some transmission peaks appear at some certain angles, which correspond to the DP. When d A d B 20nm(25nm) , a transmission peak located at the incidence angle 0 =0 , which corresponds to the figure 2(d) (2(f)). When the lattice constants are d A d B 40nm , a pair of tunneling states with unity transmission probability appear in the transmission spectrum, which means extra DPs appear in the band structure, as shown in figure 4(b). Furthermore, the difference of electronic band gaps between the out-bound form (figure
1(a)) and in-bound form (figure 1(b)) is presented in figure 6. We take d A, B 20nm in figures 6(a) and (c), while d A, B 20 3nm in figure 6(b), d A, B 20 9nm in figure 6(d). It can be seen from the figure that the location of new DP is independent of the lattice constants with the same order n , and the electronic band structure of out-bound form is more complex than that of in-bound form. It is notable that the extra DPs located at k y 0 can also emerge in the 3-th Cantor GSLs in out-bound form, as shown in figure 6(a), where the locations of additional DPs are different from the DP in the case of normal incidence. It indicates that the multi-DP can be controlled by adjusting the order n (see equation (10) or (11)), which provides the flexibility to control the electron transport in Cantor GSLs by adjusting the order. The conductance and the Fano factor in the 2-th Cantor GSL, (ABABBBAAA)8, is presented in figure 7. It shows that the angular-averaged conductance curve reaches its minimum at the DP, and forms a linear cone around the DP. Meanwhile, the Fano factor reaches the value of 1/3 approximately at the corresponding DP’s location or conductance minimum. Moreover, from figure 7, we can see that the conducatance and Fano factor shift with the ratio d B d A , which indicates that the conductance and Fano factor of Cantor GSLs can be modulated by adjusting the ratio of lattice constants. In addition, we should point out that the trace map and corresponding electronic transport properties in Cantor GSLs are different from the Fibonacci, Thue-Morse, and double-periodic case. The ratio N A N B in Thue-Morse sequence is always equal to unit, while it depends on the variable order n for Fibonacci and double-periodic case, as well as the Cantor sequence. However, the position of new DP for Fibonacci GSLs will become stable for
n 1 since limn N A N B 1 5
2 while limn N A N B 2 for double-periodic
sequence. It is interesting that limn N A N B 0 for the Cantor case, thus the location of new DP in Cantor sequence tends to zero for n 1 . 4 Summary In summary, we have investigated the electronic band gaps and transport in the
graphene-based Cantor superlattices with the transfer matrix method. It is found that the
zero k gap associated with the DP is robust against the lattice constants and less sensitive to the incidence angle, which can be applicable to control the electron transport in Cantor GSLs. The position of zero k gap in Cantor GSLs shifts to lower energy region with the increase of order n , and tends to zero for n 1 , which is different from the periodic, Fibonacci, Thue-Morse and double-periodic GSLs. Furthermore, multi-DPs can be controlled by adjusting the lattice constants and the order n , and an expression for their location is derived. We hope the results obtained in such fractal structure will have applications in various graphene-based electron devices, such as graphene-based electronic omnidirectional reflector and filters. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant Nos. 11204170, 61108010).
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Figures
Fig. 1. Example of Cantor GSLs, (a) out-bound form, (b) in-bound form. The black and white blocks represent the barrier A and well B, respectively. (c) The schematic profiles of the potentials VA and VB corresponding to 2-th Cantor structure both in (a) and (b).
Fig. 2. Trace-maps for Cantor GSL as a function of order n under (a) dA dB 20nm , (b)
dA dB 25nm ,
(c)
2dA dB 40nm
with VA 50meV ,
VB 0meV
and
k y 0.01nm1 . (d), (e) and (f) are band structures for n 2 corresponding to (a), (b) and (c), respectively. The horizontal solid red line represents the location of new Dirac point, and the white areas are band gaps while the solid areas are passing bands.
Fig. 3. Transmission spectra for Cantor GSL, (ABABBBABA)8 , with (a) different widths of barriers and wells at 0 10 . Here, solid black line for dA dB 15nm , dotted blue line for dA dB 20nm , and dashed-dotted red line for 2dA dB 40nm ; (b) different incidence angles, 0 10
(solid black line), 15
(blue dotted blue line) and 20
(dashed-dotted red line) with the lattice constant dA dB 20nm . Other parameters are the same as those in figure 2.
Fig. 4. (a) Band gaps depending on the lattice constant d with k y 0.01nm1 , here
dA dB d . (b) Electronic band structures for dA dB 40nm . The order is n 2 , other parameters are the same as those in figure 2.
Fig. 5. Transmission spectrum for a S2 = ABABBBABA Cantor GSL with different incidence angle at E 22.22meV . Here, dA dB d , and the other parameters are the same as those in figure 2.
Fig. 6. Electronic band structures for out-bound form (a) and (c), and in-bound form (b) and (d). The order n 3 for (a), (b), while n 4 for (c), (d). The lattice constant in (a), (c) is
dA dB 20nm , while dA dB 20 3nm for (b), dA dB 20 9nm for (d). Other parameters are the same as those in figure 2.
Fig. 7. Conductance (a) and (b) and Fano factor (c) and (d) versus Fermi energy in Cantor GSL, (ABABBBAAA)8 . Other parameters are the same as those in Fig. 2.
Highlights The Cantor graphene superlattices are investigated theoretically. Cantor fractal structure can possess an unusual Dirac point. The Dirac corresponds to the zero-averaged wave number. Multi-Dirac-points appear by adjusting the lattice constants and the order. The electron transport in Cantor fractal structure can be controlled.