1International School for Advanced Studies, SISSA-ISAS, I-34014 Trieste, Italy2Institut für Theoretische Physik, Universität zu Köln, D-50937 Köln, Germany
3Institut für Theoretische Physik, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany�Received 8 August 2008; revised manuscript received 25 September 2008; published 26 January 2009�
We study the behavior of charge carriers in graphene in inhomogeneous perpendicular magnetic fields. Weconsider two types of one-dimensional magnetic profiles, uniform in one direction: a sequence of N magneticbarriers and a sequence of alternating magnetic barriers and wells. In both cases, we compute the transmissioncoefficient of the magnetic structure by means of the transfer-matrix formalism and the associated conductance.In the first case the structure becomes increasingly transparent upon increasing N at fixed total magnetic flux.In the second case we find strong wave-vector filtering and resonant effects. We also calculate the bandstructure of a periodic magnetic superlattice and find a wave-vector-dependent gap around zero energy.
The electronic properties of graphene1 in the presence ofinhomogeneous perpendicular magnetic fields have very re-cently attracted considerable theoretical attention.2–11 Ingraphene the charge carriers close to the Fermi points K andK� form a relativistic gas of chiral massless �Dirac-Weyl�DW�� quasiparticles with a characteristic conical spectrum.This has far-reaching consequences. For example, quasipar-ticles in graphene are able to tunnel through high and wideelectrostatic potential barriers, a phenomenon often referredto as Klein tunneling and related to their chiral nature.12
Moreover, in a uniform magnetic field, graphene exhibits anunconventional half-integer quantum Hall effect,13 which canbe understood in terms of the existence, among the relativis-tic Landau levels formed by the quasiparticles, of a zero-energy one.14
From a theoretical perspective, it is then interesting toexplore how the DW nature of the charge carriers affectstheir behavior in nonuniform magnetic fields. Such investi-gation has been started in Ref. 2, and here we generalize andexpand on it by studying several more complex geometries.
Experimentally, inhomogeneous magnetic profiles on sub-micron scales in ordinary two-dimensional electron gases�2DEGs� in semiconductor heterostructures have been pro-duced in several ways, and magnetic barriers with heights upto 1 T have been obtained. One approach exploits the fringefield produced by ferromagnetic stripes fabricated on top ofthe structure.15 Another possibility consists in applying a uni-form magnetic field to a 2DEG with a step.16 In yet anotherapproach, a film of superconducting material with the desiredpattern is deposited on top of the structure, and a uniformmagnetic field is applied.17 In this way, magnetic structureswith different geometries have been experimentally realized,and their mesoscopic transport properties have been studied,e.g., transport through single magnetic barriers18 andsuperlattices,19 magnetic edge states close to a magneticstep,20 and magnetically confined quantum dots or antidots.21
Correspondingly, there exists an extensive theoreticalliterature,22 which elucidates the basic mechanisms underly-ing the behaviors observed in experiments.
In principle, the same concepts and technologies can beused to create similar magnetic structures in graphene oncethe graphene sheet is covered by an insulating layer, whichhas recently been demonstrated feasible.23,24 Although at thetime of writing there is yet no published experimental workdemonstrating magnetic barriers in graphene, this should bewithin reach of present-day technology, which provides mo-tivation for the present work.
In a previous paper2 we showed that, in contrast to elec-trostatic barriers, a single magnetic barrier in graphene to-tally reflects an incoming electron, provided the electron en-ergy does not exceed a threshold value related to the totalmagnetic flux through the barrier. Above this threshold, thetransmission coefficient strongly depends on the incidenceangle.2,8 These observations were used to argue that chargecarriers in graphene can be confined by means of magneticbarriers, which may thus provide efficient tools to control thetransport properties in future graphene-based nanodevices.
Here we focus on more complex multiple barrier configu-rations and magnetic superlattices. We consider two types ofone-dimensional profiles. In the first case the magnetic fieldin the barrier regions is always assumed to point upward,while in the second it points alternatingly upward and down-ward. We shall see that there are sharp differences in thetransport properties of the two cases. In particular, in theup-down geometry we find a much stronger angular depen-dence of the transmission coefficient, which leads to an in-teresting wave-vector filter effect.
The outline of the paper is the following. In Sec. II weintroduce the Dirac-Weyl Hamiltonian for graphene, the twotypes of magnetic profiles we consider in the rest of thepaper, and the transfer-matrix formalism for Dirac-Weyl par-ticles. In Secs. III and IV we compute and discuss the trans-mission coefficient separately for the two cases. In Sec. V weconsider a periodic magnetic superlattice and determine itsband structure. Finally, in Sec. VI we summarize our resultsand draw our conclusions.
II. HAMILTONIAN AND TRANSFER MATRIX
Electrons in clean graphene close to the two Fermi pointsK and K� are described by two decoupled copies of the
Dirac-Weyl equation. We shall focus here on a single valleyand neglect the electron spin.25 Including the perpendicularmagnetic field via minimal coupling, the DW equation reads
vF� · �− i� � +e
cA�� = E� , �1�
where �= ��x ,�y� are Pauli matrices acting in sublatticespace and vF=8�105 m /s is the Fermi velocity ingraphene. In the Landau gauge, A= �0,A�x��, with Bz=�xA,the y component of the momentum is a constant of motion,and the spinor wave function can be written as ��x ,y�=��x�eikyy, whereby Eq. �1� is reduced to a one-dimensionalproblem,
� − E − i�x − i�ky + A�x��− i�x + i�ky + A�x�� − E
�� = 0. �2�
Equation �2� is written in dimensionless units: with B thetypical magnitude of the magnetic field, and �B=��c /eB theassociated magnetic length, we express the vector potentialA�x� in units of B�B, the energy E in units of �vF /�B, and xand ky, respectively, in units of �B and �B
−1. The values oflocal magnetic fields in the barrier structures produced byferromagnetic stripes range up to 1 T, with typical values ofthe order of tenth of tesla. For B�0.1 T, we find �B�80 nm and �vF /�B�7 meV, which set the typical lengthand energy scales.
We shall consider two types of magnetic field profiles. Inthe first case, illustrated in Fig. 1, the profile consists of asequence of N magnetic barriers of equal height B �assumedpositive for definiteness� and width dB, separated by non-magnetic regions of width d0. The vector potential is thenchosen as
A�x� = 0, x � �− �,0�ndB + �x − xn� , x � �xn,xn + dB��n + 1�dB, x � �xn + dB,xn+1�NdB, x � �xN,�� ,
�3�
where n=0, . . . ,N−1 and xn=n�d0+dB�. The quantity NdB isthe total magnetic flux through the structure per unit length
in the y direction.26 We shall refer to this profile as the mul-tiple barriers case and discuss it in Sec. III.
In the second case, illustrated in Fig. 2, each magneticbarrier is followed by a region of width d−B of opposite mag-netic field. The vector potential is accordingly chosen as
A�x� = 0, x � �− �,0�nD + �x − xn� , x � �xn,xn + dB�nD + �2dB + xn − x� , x � �xn + dB,xn+1�ND , x � �xN,�� ,
�4�
where n=0, . . . ,N−1, xn=n�dB+d−B�, and D=dB−d−B. Weshall refer to this profile as the alternating barrier-well caseand discuss it in Sec. IV. The parameter D has the meaningof net magnetic flux through a cell formed by a barrier and awell. For D=0 this profile can be extended to a periodicmagnetic superlattice, a case considered in Sec. V.
With our gauge choice, the value of the vector potentialon the right of the structure is equal to the total magnetic flux� through it
� � A�x xN� = �NdB, case 1
N�dB − d−B� , case 2, �5�
which is an important control parameter for the transportproperties. In both cases, the solutions to Eq. �2� can beobtained by first writing the general solution in each regionof constant Bz as linear combination �with complex coeffi-cients� of the two independent elementary solutions, and thenimposing the continuity of the wave function at the interfacesbetween regions of different Bz to fix the complex coeffi-cients. This procedure is most conveniently performed in thetransfer-matrix formalism. Here we directly use this ap-proach and refer the reader to Refs. 27 and 28 for a detaileddiscussion.
The transfer matrix,
T̂ = �T11 T12
T21 T22� , �6�
relates the wave function on the left side of the magneticstructure �xx0=0�,
0d dB
A
B
x x x x x0 1 2 3 N
FIG. 1. Magnetic profile Eq. �3�: N magnetic barriers of widthdB separated by nonmagnetic regions of width d0.
d−B
dB
A
B
xx x x x0 1 2 3 N
FIG. 2. Magnetic profile Eq. �4�: N magnetic barriers of widthdB separated by magnetic wells of width d−B.
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��x� = � 1
kxi + iky
E�eikx
i x + r� 1
− kxi + iky
E�e−ikx
i x, �7�
where kxi =�E2−ky
2, to the wave function on the right side�xxN�,
��x� = t�kxi
kxf � 1
kxf + i�ky + ��
E�eikx
fx, �8�
where kxf =�E2− �ky +��2. The coefficients r and t are, re-
spectively, the reflection and transmission amplitudes, andwe used that with our gauge choices �Eqs. �3� and �4��, thevector potential vanishes on the left of the magnetic structureand on the right side is equal to �. As usual, the factor�kx
i /kxf ensures proper normalization of the probability cur-
rent. The relation which expresses the continuity of the wavefunction is then given by29,30
�1
r� = T̂��kx
i /kxf t
0� . �9�
Solving Eq. �9� for t, we get the transmission probability T as
T�E,ky� = �t�2 =kx
f
kxi
1
�T11�2. �10�
Once T�E ,ky� is known, it is straightforward to compute thezero-temperature conductance by integrating T over one-halfof the �Fermi� energy surface,31
G�E� = G0�−�/2
�/2
d� cos � T�E,E sin �� , �11�
where � is the incidence angle �we measure angles withrespect to the x direction�, defined by ky =E sin � and G0=2e2ELy /�h. Ly is the length of the graphene sample in they direction and G0 includes a factor 4 coming from the spinand valley degeneracies.
Before proceeding with the calculations, we can derive asimple and general condition for a nonvanishing transmis-sion. For this purpose it is convenient to parametrize themomenta in the leftmost and rightmost regions, respectively,in terms of incidence and emergence angles, � and � f,
kxi = E cos �, ky = E sin � , �12�
kxf = E cos � f, ky = E sin � f − � . �13�
The emergence angle is then fixed by the conservation of ky,
sin � = sin � f −�
E. �14�
Equation �14� implies that transmission through the structureis only possible if �i satisfies the condition
�sin � +�
E� 1. �15�
This condition, already discussed in Ref. 2 for the case of asingle barrier, is in fact completely general and independent
of the detailed form of the magnetic field profile. It onlyrequires that the magnetic field vanishes outside a finite re-gion of space. For �� /E�2, it implies that the magneticstructure completely reflects both quasiparticles and quasi-holes. As a consequence of this angular threshold, the con-ductance has an upper bound given by
Gs�E� � G0�2 − ��
E����2�E� − ���� , �16�
with the Heaviside step function �. If the vector potentialprofile is monotonous, Gs also coincides with the classicalconductance, obtained by setting T=��1− �sin �+� /E��. If,however, A�x� is not monotonous, the classical conductanceis obtained by replacing ��� in Eq. �16� with the maximalvalue of �A� in the structure since a classical particle is totallyreflected as soon as �A�max �rather than the total flux ����exceeds twice the energy.
Before moving to Sec. III, we notice, as an aside remark,that Eq. �2� can easily be solved in closed form for E=0. Thezero-energy spinors are then given by
�+ � �1
0�ekyx+�xA�x��dx�, �17�
�− � �0
1�e−kyx−�xA�x��dx�. �18�
These wave functions are admissible if and only if they arenormalizable, which depends on the sign of ky and the be-havior of the magnetic field at x→ ��. In fact, for any A�x�,at most one among �+ and �− is admissible. If the magneticfield vanishes outside a finite region of space, as in our case,one can always choose a gauge in which A�x�=0 on the leftof the magnetic region and A�x�=� on its right. It is thenstraightforward to check that for 0ky −�, the only nor-malizable solution is Eq. �17�, whereas for −�ky 0, thenormalizable solution is Eq. �18�. In particular, we find thatwhen the net magnetic flux through the structure vanishes,there exist no zero-energy states. This is nicely confirmed bythe calculation of the spectrum of the periodic magnetic su-perlattice in Sec. V. The zero-energy state is a bound statelocalized in the structure. Additional bound states of higherenergy may also occur,8 but we do not further investigate thisproblem here.
III. MULTIPLE BARRIERS
In this section we focus on the magnetic profile in Eq. �3�.In order to compute the transfer matrix T̂, we need the twoelementary solutions of the DW �Eq. �2�� for Bz=0 and thetwo for Bz=1.
We can then construct the 2�2 matrices W0�x� andWB�x�, whose columns are given by the spinor solutions. Inthe nonmagnetic regions we have
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W0�x� = �eikxx e−ikxx
kx + i�ky + A�E
eikxx − kx + i�ky + A�E
e−ikxx� ,
�19�
where kx�x�=�E2− �ky +A�x��2. In the regions with Bz=1 wehave
WB�x� = �Dp�q� Dp�− q�
i�2
EDp+1�q�
− i�2
EDp+1�− q� � , �20�
where q=�2�A�x�+ky�, p=E2 /2−1, and Dp�q� is the para-bolic cylinder function.32 These matrices play the role ofpartial transfer matrices and allow us to express the conditionof continuity of the wave function at each interface betweenthe nonmagnetic and the magnetic regions. After straightfor-ward algebra, we get
T̂ = T̂0T̂1 . . . T̂N−1, �21�
where
T̂n = W0−1�xn�WB�xn�WB
−1�xn + dB�W0�xn + dB� �22�
is the transfer matrix29 across the �n+1�th barrier, and weremind that xn=n�d0+dB�.
From Eqs. �10�, �21�, and �22� we numerically evaluatedthe transmission probability T for various sets of parameters.The results are illustrated in Figs. 3, 5, and 7. Figure 3 showsthe angular dependence of the transmission coefficient atfixed energy for several values of N, but keeping constant themagnetic flux � through the structure. In agreement with thediscussion in Sec. II and Eq. �15�, we observe that the rangeof angles where T�0 remains the same, � 0, upon increas-ing the number of barriers. At the same time, however, thetransmission itself is modified, and oscillations appear,whose number increases with N and for larger separationsbetween the barriers.
More remarkably we find that rarefying the magnetic fieldby adding more barriers without changing the total flux �,the transmission probability approaches the classical limit,where it is zero or one depending on whether the incidenceangle exceeds or not the angular threshold; see Fig. 3. Cor-respondingly, the conductance as a function of N approachesthe classical limit �Eq. �16��; see Fig. 4. As expected, thesame limit is also approached upon increasing the energyespecially for large N. This is clearly illustrated in Fig. 5 forthe transmission, and in Fig. 6 for the conductance: one sees
10.50.51
N=1N=2N=3N=10
FIG. 3. �Color online� Angular dependence of the transmissionprobability at E=1, through N=1,2 ,3 ,10 barriers of width dB
=1 /N �keeping in this way constant the flux �=NdB=1� andspaced by d0=10.
0.7
0.75
0.8
0.85
0.9
0.95
1
1 3 5 7 9 11 13 15
G/G
0
N
FIG. 4. The conductance G /G0 at E=1 for several values of Nwith dB=1 /N, such that �=1, and d0=10 �black dots linked bylong-dashed line, which is a guide for the eyes only�. The upperbound �dashed line� corresponds to the classical value Gs /G0=1�see Eq. �16�� while the dotted line is the curve given by Eq. �26� asa function of N.
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E=3.2E=3.4E=4E=5
FIG. 5. �Color online� Angular dependence of the transmissionprobability, for different values of E, fixing dB=1, d0=10, and N=6.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
G/G
0
E
FIG. 6. The conductance as a function of the energy for N=6barriers with dB=1 and d0=10 �solid line�. The dashed line is thecurve given by the classical limit Eq. �16�.
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that already for six barriers the classical limit provides a verygood approximation. The classical limit is instead hardlyachieved changing dB, except when dB is close to the ex-treme values 0 and E /N; see Figs. 7 and 8.
In conclusion the main result of this section is that, atfixed flux �, the larger is the number of barriers, the moretransparent is the magnetic structure. This is a purelyquantum-mechanical effect, peculiar to magnetic barriers.
Qualitatively this behavior can be explained as follows.To calculate the probability for a relativistic particle to gothrough a very thin single barrier, we can simulate the profileof A with a step function with height � /N in order to havethe same flux of a magnetic barrier with width dB=� /N. Inthis case the transfer matrix is simply
T̂ = W0�0−�−1W0�0+� , �23�
where W0�0−� is given by Eq. �19� with A=0 and kx=kxi ,
while in W0�0+� we have A=� /N and consequently kx=kxf .
From Eq. �10� we get the following transmission probabilityfor a single barrier:
T1��� �4 cos � cos � f ��1 − �sin � + �/NE��
�cos � + cos � f�2 + ��/�NE��2 , �24�
where � f is defined by E sin � f =E sin �+� /N. For N bar-riers we can roughly estimate the probability for the particleto cross the magnetic structure to be
TN��� � T1���N��1 − �sin � + �/E�� , �25�
where we have put by hand the global constraint of momen-tum conservation. Using the expression above we can thencalculate the conductance applying Eq. �11�. To simplify thecalculation, in order to have a qualitative description of theconductance behavior, we further approximate cos � f�cos � in Eq. �24�, valid for small � /N, and replace cos2 �with its average 1/2, getting at the end an approximated ex-pression for G which reads
G � Gs� 2N2E2
2N2E2 + �2�N
, �26�
being � the total flux for N barriers. In Fig. 4 we comparethe exact calculation with the approximate one given by Eq.�26� and find a surprisingly good agreement. The small dis-crepancy has various possible sources: the angular depen-dence of the transmission, which is averaged out in the ap-proximate calculation; the finite width of the barriers; and thefinite separations among the barriers, which may let the par-ticles bounce back and forth, in this way reducing the trans-mission. This latter effect, therefore, suppresses a bit the con-ductance predicted by Eq. �26�.
IV. ALTERNATING MAGNETIC BARRIERS AND WELLS
Next, we consider the magnetic profile in Eq. �4�, illus-trated in Fig. 2. In order to construct the transfer matrix inthis case we need WB and the partial transfer matrix for theregions with Bz=−B, which is given by
W−B�x� = �Dp+1�− q� Dp+1�q�
− i�2
E�p + 1�Dp�− q�
i�2
E�p + 1�Dp�q� � .
�27�
After some algebra, we then get
T̂ = W0−1�x0�W−B�x0�T̂0T̂1 . . . T̂N−2T̂N−1W−B
−1�xN�W0�xN� ,
�28�
where
T̂n = W−B−1�xn�WB�xn�WB
−1�xn + dB�W−B�xn + dB� �29�
is the transfer matrix29 across the �n+1�th magnetic barrierand xn=n�dB+d−B�. Note that Eq. �29� differs from Eq. �22�since now on the right and on the left of a magnetic barrierthere is a magnetic well rather than a nonmagnetic region. As
in Sec. III, the numerical evaluation of T̂ is straightforward,and the results for the transmission probability and for theconductance are illustrated in Figs. 9–15.
Figure 9 shows the angular dependence of T for a singleblock consisting of a barrier followed by a well of differentwidth. The plot emphasizes the very strong wave-vector de-pendence of the transmission and shows that by tuning dBand d−B one can achieve very narrow transmitted beams.This suggests an interesting application of this structure as amagnetic filter, where only quasiparticles incident with an
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dB=0.8dB=1dB=1.1dB=1.2
FIG. 7. �Color online� Angular dependence of the transmissionprobability, for different values of dB, fixing E=2, d0=10, and N=3.
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1 1.2
G/G
0
dB
FIG. 8. The conductance as a function of dB for N=3 barrierswith E=2 and d0=10 �solid line�. The dashed line is the curve givenby the classical limit Eq. �16�.
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angle within a very small range are transmitted.More explicitly, in order to select a narrow beam at an
angle 0��2 �at fixed energy�, one can choose the widths
such that
dB
E= �1 − sin �� − �1, �30�
d−B
E= 2 − �2, �31�
with �1,2�1. With this choice, � is very close to the angularthreshold for the first barrier, and the well is close to betotally reflecting. The combination of these two effects leadsto the narrow beam. For − �
2 �0 it is enough to flip themagnetic field. These relations hold better at small values of���. In the rest of this section, we focus on structures withdB=d−B, i.e., when the total magnetic flux through the struc-ture vanishes.
For several blocks of barriers and wells we observe aninteresting recursive effect shown in Fig. 10. There areangles at which, upon increasing N, the transmission takes atmost Nm values, where Nm is the smallest number of blocksfor which the transmission is perfect �i.e., T=1�. Some ofthese angles are emphasized in Fig. 10 by dashed lines. Atthose angles, for instance, magnetic structures with N equalto integer multiples of 2, 3, and 5 exhibit perfect transmis-sion even for small energy.
These effects can be understood as follows. Suppose that,at a given angle, the transmission probability through Nmcells has a resonance and reach the value 1. Then, for anysequence consisting of a number of cells equal to an integermultiple of Nm, the transmission is 1 again. �It is crucial herethat since the magnetic flux through each block is zero, theemergence angle always coincides with the incidence angle.�At such an angle, then, T can only take, upon changing N, atmost Nm values. Notice that perfect transmission occurs alsofor low energy of the incident particle, and the angularspreading of perfect transmission is also reduced by addingmultiple blocks.
This effect in the transmission also reflects in the conduc-tance. Figure 11 shows that, for a particular set of param-eters, G oscillates as a function of N with period 3. However,for a different value of dB, shown in the inset, the period is 2.This unexpectedly strong dependence of the conductance onadding or removing blocks of barriers and wells could beexploited to design a magnetic switch for charge carriers ingraphene. Moreover, we observe that the angular dependenceof the transmission is abruptly modified also by changing theenergy E of the incident particles �see Fig. 12� or the widthof the barriers dB �Fig. 14� where we observe pronouncedresonance effects. As a consequence the conductance exhib-its a modulated profile as a function of both the energy andthe barrier’s width, as illustrated, respectively, in Figs. 13and 15.
10.50.51
dB=5.8dB=2.9dB=0.8dB=0
FIG. 9. �Color online� Angular dependence of the transmissionprobability at E=6 through a structure consisting of one magneticbarrier of width dB and one magnetic well of width d−B. We fixd−B=11.7�2E and vary dB from 0 to 5.8�E.
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N=1N=2N=3N=5N=6N=10
FIG. 10. �Color online� Angular dependence of the transmissionprobability at E=1 and dB=d−B=1 for several values of N. Theblack dashed lines correspond to the angles for recursive transmis-sion of multiplicity 2 �at ��� /18�, 3 �at �=−� /6�, and 5 �at ��−� /54 and ��−7� /18�.
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1 4 7 10 13 16
G/G
0
N
1.1
1.2
1.3
1.4
FIG. 11. The conductance G /G0 at E=1 varying N with dB
=d−B=1. In the inset dB=d−B=0.8 in the same range of N.
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E=1E=1.8E=3
FIG. 12. �Color online� Angular dependence of the transmissionprobability for several values of the energy E, for dB=d−B=1 andN=6.
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V. PERIODIC MAGNETIC SUPERLATTICE
We now focus on the case of a periodic magnetic super-lattice. We observe that, if dB=d−B, the profile �Eq. �4�� canbe extended to a periodic profile, illustrated in Fig. 16, wherethe elementary unit is given by the block formed by a barrierand a well. Imposing periodic boundary conditions on thewave function after a length L=xN=N� �where �=2dB�, i.e.,��x0�=��xN�, and defining the matrix,
� = WB−1�0�W−B�0�W−B
−1�dB�WB�dB� , �32�
standard calculations27 lead to the quantization condition forthe energy,
2 cos�Kx�� = Tr � . �33�
At fixed ky, Eq. �33� gives the energy as a function of theBloch momentum Kx= 2�n
L . Notice that Kx is related to theperiodicity of the structure and parametrizes the spectrum. Itshould not be confused with the x component of the momen-tum kx used in Secs. II–IV. Figure 17 illustrates the first twobands as a function of Kx for two values of ky. Figure 18shows the contour plot for Tr � as a function of E and ky.We find two interesting main features. First, around zero en-ergy there is a gap, whose width decreases for larger valuesof �ky�. This is in agreement with the fact that for a magneticprofile with zero total flux there exist no zero-energy states.
Second, for some values of ky, the group velocity vy = �E�ky
diverges �see Fig. 18 close to �ky� �E��0.3�. The propertyof superluminal velocity has already been observed formassless Klein-Gordon bosons in a periodic scalarpotential.28 In this work, such property has been observed formassless Dirac-Weyl fermions in a periodic vector potential.
VI. SUMMARY AND CONCLUSIONS
In this paper we have studied the transmission of chargecarriers in graphene through complex magnetic structuresconsisting of several magnetic barriers and wells and therelated transport properties. We focused on two differenttypes of magnetic profiles. In the case of a sequence of mag-netic barriers, we have found that the transparency of thestructure is enhanced when the same total magnetic flux isdistributed over an increasing number N of barriers. Thetransmission probability and the conductance then approachthe classical limit for large N; see in particular Figs. 3 and 4.
The behavior of alternating barriers and wells turns out tobe even more interesting. We have shown that a single unitconsisting of a barrier and a well of suitable widths can beused as a very efficient wave-vector filter for Dirac-Weylquasiparticles; see Fig. 9. With several blocks we have ob-
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 1 2 3 4 5 6 7 8
G/G
0
E
FIG. 13. The conductance G /G0 as a function of the energy Efor N=6 with dB=d−B=1 �solid line�. The dashed line is the classi-cal limit, i.e., �2−dB /E���2E−dB�.
10.50.51
dB=0.8dB=1dB=1.1dB=1.2
FIG. 14. �Color online� Angular dependence of the transmissionprobability for several values of dB=d−B with N=3 and E=1.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 0.5 1 1.5 2
G/G
0
dB
FIG. 15. The conductance G /G0 as a function of dB=d−B forN=3 with E=1. The dashed line is the classical limit.
dB
d−B
B
A
x x x x0 1 2 3 xN
FIG. 16. Periodic superlattice of alternating magnetic barriersand wells with dB=d−B.
MULTIPLE MAGNETIC BARRIERS IN GRAPHENE PHYSICAL REVIEW B 79, 045420 �2009�
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served strong resonant effects, such that at given angles onegets narrow beams perfectly transmitted even for low energyof the incident quasiparticles; see Fig. 10. As a result, theconductance is drastically modified by adding or removingblocks; see Fig. 4. This suggests possible applications asmagnetic switches for charge carriers in graphene.
Since most of our results concern the angular dependenceof the transmission coefficient, a comment on the possibilityto measure it is in order here. Such measurement should, inprinciple, be feasible, following for example ideas put for-ward in Ref. 33. The idea consists in using a small electriccontact �for example, the tip of a scanning tunneling micro-scope �STM�� to inject electrons in the graphene sheet onone side of the magnetic structure, and a second local contacton the other side as a probe to detect the transmitted elec-trons. The current measured between the two contacts wouldthen reflect the transmission probability through the struc-ture. If the contacts are made smaller than the de Brogliewavelength of the electrons �which is large for small doping,the situation considered here�, then one may be able to re-solve the spatial structure of the local current distribution and
thus get information on the angular dependence of T. Wehope that our paper will further stimulate experimental workon the rich physics of magnetic structures in graphene.
ACKNOWLEDGMENTS
We thank R. Egger and W. Häusler for several valuablediscussions. This work was supported by the MIUR project“Quantum noise in mesoscopic systems,” by the SFBTransregio 12 of the DFG, and by the ESF networkINSTANS.
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-2
-1
0
1
2
-3 -2 -1 0 1 2 3
E
Kx
FIG. 17. The spectrum for the periodic superlattice in Fig. 16with dB=d−B=1 at ky =0 �full points� and ky =1 �cross poins�. Kx isthe Bloch momentum.
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