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since the successful fabricating of graphene in 2004 ,  @xcite its excellent transport properties ,  @xcite such as the exceptional electron and thermal transport properties at room temperature , have attracted a lot of research interest to apply it in making future electronic devices . @xcite many ideas and attempts have been going on . one of the most fundamental and revolutionary attempts is to control the valley degree of freedom to realize the signal s storage and transport.@xcite supposed an achievement of this idea in future , graphene will probably replace all semiconductors in conventional electronics and bring a new era in many fields . before the coming of the new era , there is a long way to improve the control over transport in graphene .    inspired by the crucial role of the doping in semiconductor , many researchers have studied the effects related to disorder in graphene,@xcite which consists of adsorbed atoms ( or molecules ) , charged impurities , vacancies , or other topological defects . it is anticipated to modify the transport property of pristine graphene to fulfill the requirement . recently , a peculiar topological line - defect in graphene was reported experimentally by lahiri et al . @xcite this topological line - defect is created by alternating the stone - thrower - wales defect and divacancies , leading to a pattern of repeating paired pentagons and octagons , as shown in fig . 1 . in particular , it is found in this experiment that this line - defect has metallic characteristics . great attention has been attracted to the line - defect and further investigations manifest a variety of its promising applications.@xcite    although a lot of theoretic proposals are presented in graphene to realize the valley filter before , all of them are not realized till now due to the extreme difficulty in the experimental implementation . after the discover of line - defect , gunlycke and white  @xcite reported that a valley polarization near @xmath0 can be achieved by scattering off a line - defect.@xcite this original idea subsequently arouse physics community s intensive interests.@xcite as a new discovery , the extended line - defect is very promising to realize the valley polarization in graphene . however , because the @xmath0 valley polarization only maintains at a large incident angle in these reported results.@xcite a serious problem will be encountered in experiment that the electron must always follow the direction of the line - defect to maintain a high valley polarization . this transport characteristic is almost fatal to utilize efficiently the high valley polarization . the purpose of this paper is to address a feasible approach to realize a @xmath0 valley polarization at a small incident angle by using a resonator , which consists of several paralleling line - defects .    given a high valley polarization by scattering off one line - defect , a question is putted forward naturally : what will happen in the presence of two or more line - defects ? motivated by finding an efficient realization of the high valley polarization , we therefore study the electron transmission coefficient in the presence of several line - defects . intriguingly , it is found that an obvious change happens to the transmission coefficients with varying the distance between two line - defects . in particular , for the electron with a definite momentum and incident angle , the transmission coefficients can reach to one at a definite distance between two line - defects . simple numerical analysis argues that the large transmission coefficients should be attributed to the standing wave structure due to the electron s quantum interference , which is also named as fabry - p@xmath1rot interference in optics . based on a further numerical simulation , an efficient device to realize a @xmath0 valley polarization is further proposed for six line - defect configuration at the end . this fascinating finding opens a way to realize the valley polarization by line - defects . the rest of this paper is organized as follows : in sec . [ sec : models ] , we introduce the hamiltonian in the tight - binding model and present the formula of the transmission coefficient . the numerical results are discussed in sec . [ sec : discussions ] . finally , a brief summary is given in sec . [ sec : conclusions ] . in the tight - binding approximation , a single layer of graphene with a line - defect , sketched in fig . 1(a ) , can be described by the following hamiltonian : @xmath2 where @xmath3 and @xmath4 represent the electron annihilation operators on the graphene site @xmath5 and line - defect sites , respectively , and @xmath6 refers to the nearest neighboring sites . supposed a variation less than @xmath7 in the hopping terms @xmath8 and @xmath9,@xcite here we adopt @xmath10 . is from the fourier transformation along y axis . ]    because of the existence of line - defect , the translation symmetry along @xmath11 axis breaks down for a graphene system , as sketched in fig . however , the translation symmetry along @xmath12 axis still retains and therefore @xmath13 is a good quantum number , namely a conserved quantity . for the sake of the convenience in computing , we do the fourier transform only along @xmath12 axis as follows : @xmath14 @xmath15 represents the lattice constant in pristine graphene . after the fourier transformation , the hamiltonian in eq . ( [ m1 ] ) is decoupled into @xmath16 . note that here @xmath17 can be effectively represented by a quasi - one - dimensional lattice model , as shown in fig . readily , we can describe the hamiltonian @xmath17 as follows : @xmath18 where @xmath19 , @xmath20 $ ] , and @xmath21 represents the unit length between the neighboring supercells at the graphene part . here , @xmath22 represent the position of a supercell , @xmath23 takes the integer number from 1 to 4 to denote the different columns in a supercell , and @xmath24 in @xmath25 corresponds to the up / down site in the same column in fig . 1(c ) .     or @xmath26 dirac cone . ( b ) the relationship between the electron s incident angle and momentums . ( c ) and ( d ) manifest the transmission coefficient in @xmath27 and @xmath26 valleys . in this case , there is only one line - defect . here , the dotted line represents the approximate result in the low energy limit : @xmath28    according to the landauer - bttiker formula and the hamiltonian in eq . ( [ m3 ] ) , the transmission coefficient of the line - defect system at zero temperature and low bias voltage can be represented as :  @xcite @xmath29},\end{aligned}\ ] ] where @xmath30 is the retarded / advanced green s function related to the line - defect hamiltonian and @xmath31 with the retarded / advanced self - energy @xmath32 . note that here the left / right lead is represented by a semi - infinite quasi - one - dimensional graphene lattice sketched in fig . 1(c ) and the sample hamiltonian refers to the line - defect hamiltonian , namely @xmath33 . ( b ) and @xmath26 ( c ) are mapped as a function of the incident angle @xmath23 in the presence of two line - defects . here , electron s energy is set as @xmath34 . ] it is worth noting that we can readily compute the transmission coefficients of the electron with a fixed incident angle @xmath23 in two dirac points , @xmath27 and @xmath26 , by using the above formula . as shown in fig . 1(b ) , two dirac points @xmath27 and @xmath26 which are located at @xmath35 $ ] for a pristine graphene are now cast at @xmath36 and @xmath37 for a graphene with a line - defect , @xmath38 $ ] . therefore , an electron in two valleys satisfies the following relations : @xmath39 and @xmath40 , where @xmath41 represents electron s group velocity along the @xmath42 direction . combined with the dirac electron s linear dispersion relation @xmath43 , the transmission coefficients can be readily mapped as a function of the electron incident angle @xmath23,@xcite where @xmath44 . for simplicity in discussing , we will never distinguish the difference of the dirac points in two brillouin zones in the following and uniformly adopt @xmath27 and @xmath26 to represent the two nonequivalent dirac points . besides , some previous papers reported that in the vicinity of the dirac point , the transmission coefficient of low energy electron will obey the law : @xmath45 . in order to make this work essentially self - contained , we also confirm this conclusion and present the related derivations in appendix . even though the transmission coefficient in the vicinity of the dirac point was given by an approximated method in many references,@xcite to the best of our knowledge , the exact numerical calculation without any analytical approximations is not unambiguously shown before . given the realization of the valley polarization using the line - defect in future , what the transmission coefficient behaviors at a relatively large energy is a fundamental question .    in fig . 2 , the transmission coefficient is mapped as a function of the electron s incident angle for different energies . it can be observed the line of @xmath46 ( red ) overlaps exactly with the dotted line , @xmath47 , which represents the transmission coefficient of low energy electron in the vicinity of the dirac point by an approximate method and is also given in appendix . one subtle difference between the numerical and approximate results is zero transmission probability nearby @xmath48 for exact numerical result . however , this physical observation is not manifested in the approximate result ( dotted line ) . a severe deviation from the dotted line is clearly seen for the lines with @xmath49 , which indicates that a excellent valley transmission can only be maintained at not very large energy . ( left ) and @xmath26 ( right ) valleys . in this case , two line - defects are considered and the fixed distance between them is @xmath50 . ] the search of the valley polarization material or device attracts physics community greatly . although a proposal was given recently by scattering off a line - defect,@xcite it looks much difficult for graphene with only one line - defect to make electron leave away from the line - defect and finally reach the key device , and meanwhile maintain at a high valley polarization in this process . inspired by the valley polarization in the line - defect graphene , we expect this problem can be resolved by scattering off two or several line - defects .    in fig . 3 , the transmission behavior is mapped as a function of the incident angle in the presence of two line - defects . 3(a ) shows a lattice model of graphene with two line - defects and meanwhile presents a schematic interpretation of @xmath51 which represents the distance between two nearest neighbouring line - defects . using the formula ( 4 ) , the transmission coefficients in the @xmath27 and @xmath26 valleys are mapped as a function of the incident angle in fig 3(b ) and 3(c ) . a mirror symmetry with respect to @xmath52 can be seen clearly between them . comparing to the case of one line - defect , there is almost no distinct change for the lines of @xmath53 both in fig . 3(b ) and 3(c ) . however , the transmission coefficient starts varying for @xmath54 and in particular some transmission peaks can be observed obviously at small incident angles for the lines of @xmath55 and @xmath56 . this exciting finding makes us fascinated to explore what happens to the two - line - defect graphene and why there is a high transmission probability at some incident angles .     which is exactly the green circle path in fig . 4 . two lines ( blue and red ) correspond to that of @xmath27 and @xmath26 valleys , respectively . in this case , two line - defects are considered and the fixed distance between them is @xmath50 . ]    in order to demonstrate a whole physical picture , we map the transmission coefficient as the function of @xmath57 in fig . the left and right maps , which correspond to the @xmath27 and @xmath26 valleys , respectively , manifest a clear symmetry . intriguingly , two large transmission regions ( dark red ) can be observed both in the @xmath27 and @xmath26 valleys . note that the positive ( negative ) @xmath58 represents an incident electron along positive x direction ( negative x direction ) . these peaks remind us some quantum interferences occur in the presence of two line - defects , e.g. electron s resonant tunneling phenomena . to confirm the origin of the high transmission above , we plot two lines selectively in fig . these two lines which belong to @xmath27 and @xmath26 valleys respectively correspond to half circle of the green route in fig . 4 , representing @xmath59 . obviously , the transmission peaks at @xmath60 and @xmath61 are from the intersection of the green circle route and the high transmission region ( dark red ) . when substituting @xmath62 and @xmath63 into the formula : @xmath64 and @xmath65 , we find two quantities are equivalent within the physical error range . this is to say , the @xmath0 transmission does originate from the resonant tunneling , which is often observed in the system of quantum dot.@xcite     valley is given in the presence of several paralleling line - defects : ( a ) @xmath66 , ( b ) @xmath67 , ( c ) @xmath68 , and ( d ) @xmath69 . note that here @xmath70 represents the number of paralleling line - defects . in each case , four lines with differen @xmath71 , the distance of two nearest neighbouring line - defects , are manifested . here , electron s energy is set as @xmath34 ]    according to the quantum theory of wave - particle duality , the low energy electron in the vicinity of the dirac point is inclined to exhibit undulation characteristics . when scattering off two line - defects , a standing wave state between two line - defects helps electron resonantly tunnel through the barriers of the line - defects . consequently , we can observe this perfect transmission phenomenon in above figures . using the valley polarization formula @xmath73 , the @xmath0 valley polarization can not be realized at very small incident angle in the presence of two line - defects , e.g. at the positions of the @xmath60 and @xmath61 peaks . nevertheless , an exciting thing is that @xmath0 valley polarization can almost be reached when @xmath23 takes the value of @xmath74 $ ] , a relatively small incident angle . at least , the results argue that the high valley transmission at a small incident angle is feasible by scattering off two line - defects . ( left ) and @xmath26 ( right ) valleys . in this case , six line - defects are considered and the fixed distance between arbitrary two neighboring line - defects is @xmath75 . ]      inspired by the above results , we expect an excellent case could occur to the graphene with several line - defects . in fig . 6 , the transmission coefficient only in @xmath27 valley is shown in the presence of several paralleling line - defects : ( a ) @xmath66 , ( b ) @xmath67 , ( c ) @xmath68 , and ( d ) @xmath69 , where @xmath70 represents the number of paralleling line - defects . in fig . 6(a ) with @xmath66 , almost no obvious change can be observed with comparison to the case of @xmath76 in fig . nevertheless , one extra transmission peak still appears in the line of @xmath55 ( red ) . without question , it should be attributed to the effect of multiple interference between line - defects . when setting @xmath67 in fig . 6(b ) , except @xmath77 , the transmission coefficients of all other lines become active and manifest some peaks at some fixed incident angles . for the cases of @xmath68 in fig . 6(c ) and @xmath69 in fig . 6(d ) , more transmission peaks appears and overlap each other . however , the electron transmission coefficient in case of @xmath77 decreases gradually with increasing the number of paralleling line - defects . it is because at @xmath34 electron s wave length is considerably larger than the distance of two nearest neighbouring line - defects , @xmath77 .    for sake of concreteness , here we restrict to the case of six line - defects and make great efforts to manifest clearly a picture of a @xmath0 valley polarization . in fig . 7 , the transmission coefficients in two valleys are mapped in the panel of @xmath78 . it is clearly shown that comparing to the case of two line - defects , more large transmission regions ( dark red ) exist in fig . 7 , the case of six line - defects . as explained in fig . 4 and 5 , these high transmission regions should likewise be from the resonant tunneling due to the quantum interference , which is also named as fabry - p@xmath1rot interference in optics .    in fig . 8 , the transmission coefficient is plotted as a function of the incident angle @xmath23 , which follows half of the green path in fig . note that here the electron momentum @xmath79 is taken as @xmath80 . obviously , the marking a , b , and c transmission peaks should be attributed to the three intersections between the green path and the large transmission regions ( dark red ) in fig . 7 . here , @xmath81 , @xmath82 , and @xmath83 illustrate that the a transmission peak is attributed to the intrinsic scattering property of a line - defect and the peaks at b and c are from the resonant tunneling of the positive and negative first standing waves , respectively .    much importantly , not only a perfectly resonant transmission but also a @xmath0 polarization can be obtained at the @xmath84 resonant peak by the polarization formula : @xmath73 . in a conclusion , we predict that the perfect @xmath0 polarization can be achieved by scattering off many line - defects , which origins in the quantum resonant tunneling phenomenon . even though only exhibiting the cases of two and six line - defects here , in principle this high transmission and thus the @xmath0 polarization can be realized in graphene with arbitrary number of line - defects . this feature shed a light on constructing a feasible valley polarization resonator by using multiple line - defects .     which is exactly the green circle path in fig . two lines ( blue and red ) correspond to that of @xmath27 and @xmath27 valleys , respectively . in this case , six line - defects are considered and the fixed distance between arbitrary two neighboring line - defects is @xmath75 . ] in summary , we study the valley polarization in the presence of several paralleling line - defects . importantly , the valley polarization changes with varying the distance between the line - defects and the number of the line - defects . in particular , the valley polarization can reach to the maximum value of one , namely a 100@xmath85 polarization , at a small incident angle . further numerical calculations manifest that @xmath0 polarization should be attributed to the behavior of electron quantum interference , more precisely the phenomenon of the standing wave between the line - defects . the findings make us get rid of helplessness in the choose of the electron s incident angle and provides a reliable way to realize a controllable valley polarization in experiment . with the great advance in graphene related technologies , a valley polarization resonator and also a detector of a valley polarization , being the key components in valleytronics , are promising to be realized in the system of the line - defect graphene .    in the preparation of this paper , we notice a reference on two paralleling line - defect in recent@xcite and intriguingly , nearly a perfect perfect valley polarization is also observed . this report also illustrates that the valley filter of several paralleling line - defects should feasible in experiment . yang liu and juntao song want to thank liwei jiang and yisong zheng for their help in the preparation of this paper . we gratefully acknowledge the financial support from nsf - china under grant nos . 11204065(j.t.s . ) , 11274089(y.l . ) , 10974043(y.x.l . ) , and 11274364(q.f.s . ) . is also supported by rfdphe - china under grant no . in this appendix , we present a detailed derivation for the transmission coefficient @xmath86 of low energy electron in the vicinity of the dirac point . note that similar derivations can also reference to some previous papers.@xcite    in the landauer - bttiker formula , eq . ( [ m4 ] ) , the green s function @xmath87 can be readily expressed as : @xmath88 . here , @xmath89 is just a two - by - two matrix related to @xmath33 and therefore the key point is to obtain the retarded self - energy function , @xmath90 . that can be solved by the surface green s function . in the following , we firstly present an analytic method of obtaining the surface green s function @xmath91 and then derive the transmission coefficient of low energy electron in the vicinity of the dirac point .    based on the schematic representation in fig . 1(c ) , a hamiltonian of a semi - infinite right lead can be written as : @xmath92 where @xmath93 means a hopping term from the @xmath94 sub - layer in the @xmath95 supercell to the @xmath96 sub - layer in the @xmath97 . obviously , the relations hold true : @xmath98 , @xmath99 , @xmath100 , and @xmath101 . thus , the eigen equation @xmath102[\mathcal{c}]=e[\mathcal{c}]$ ] can be written as : @xmath103 where @xmath104 and @xmath105 $ ] represents the eigenvalue and eigenstate , respectively . for simplicity , we cast the above eigen equation into a subspace and consequently obtain the following expression , @xmath106 here , @xmath107 where @xmath108 . note that other quantities , such as @xmath109 , @xmath110 and @xmath111 , can be also obtained by using dyson equation of green s function . for simplicity in the following derivation , we define some symbols as : @xmath112 where @xmath113 takes a form of @xmath114    following the similar derivation processes of the eigen equation above , the green s function can be written readily as : @xmath115 after a gauge transformation by a unitary matrix , which is denoted as : @xmath116 the @xmath113 matrix , all @xmath117 matrixes , and green s function in eq . ( [ a6 ] ) can be transformed into : @xmath118 where @xmath119 and @xmath120 . note that the green s function must be diagonalized if all coefficients in eq . ( [ a6 ] ) , namely all @xmath121 , can be diagonalized by this unitary transformation . there are many known mathematic method to calculate an inverse matrix of a tridiagonal matrix . however , we refrain from introducing these methods here . we use @xmath126 to denote the determinant of the tridiagonal matrix on the left side of eq . ( [ a9 ] ) and @xmath127 to denote the determinant of the tridiagonal submatrix except the former @xmath22 columns and rows of the tridiagonal matrix.@xcite then , we can obtain the recursive equations : @xmath128 and @xmath129 . in the limit of infinite long lead , corresponding to an infinite large tridiagonal matrix in eq . ( [ a9 ] ) , the relation @xmath130 maintains . from the formula @xmath131 , we can readily obtain @xmath132 where @xmath133 is adopted .    performing a inverse unitary transformation with respect to eq . ( [ a8 ] ) , the surface green s function of a semi - infinite long lead can be expressed as : @xmath134 where @xmath135 and @xmath136 . now , we get an exact expression about the surface green s function of a semi - infinite long lead .    combining the self energy function and the green s function of the line - defect @xmath137 substituting the above formula into the landauer - bttiker formula , eq . ( [ m4 ] ) , we can express the transmission coefficient exactly as : @xmath138,\end{aligned}\ ] ] where @xmath139 , @xmath140 , and @xmath141 .    in the low energy limit , it is not difficult to find @xmath142 , @xmath143 , and @xmath144 . thus , the transmission coefficient is approximately written as:@xcite @xmath145 note that we still reserve the term of @xmath146 in @xmath147 and @xmath148 because some terms in @xmath147 and @xmath148 are in the same order as @xmath146 . assuming incident electron s energy is much smaller than the hopping terms @xmath149 , @xmath8 and @xmath9 , an approximate expression of the transmission coefficient can be written as : @xmath150/2}{2\tau^2_1\tau_2\cos{\theta}(1-m_1)+e^2 + 4\tau^4_1\cos^2{\theta}e^2}\end{aligned}\ ] ] following the relations : @xmath39 , @xmath151 , @xmath152 , @xmath153 , and @xmath154 , we adopt the approximation in the low energy limit : @xmath155 , @xmath156 , and @xmath157 in eq . 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the electron transport of different conical valleys is investigated in graphene with extended line - defects . 
 intriguingly , the electron with a definite incident angle can be completely modulated into one conical valley by a resonator which consists of several paralleling line - defects . 
 the related incident angle can be controlled easily by tuning the parameters of the resonator . 
 therefore , a controllable @xmath0 valley polarization , as well as the detection of the valley polarization , can be realized conveniently by tuning the number of line - defects and the distance between two nearest neighbouring line - defects . 
 this fascinating finding opens a way to realize the valley polarization by line - defects . with the advancement of experimental technologies , this resonator is promising to be realized and thus plays a key role in graphene valleytronics .