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future device technology and quantum information processing may be based on spintronics @xcite , where one manipulates the electron s spin ( and not only its charge ) . here we address attempts to build mesoscopic spin filters ( or spin valves ) , which generate a tunable spin - polarized current out of unpolarized electron sources . much recent effort in this direction uses narrow - gap semiconductor heterostructures , where the spins are subject to the rashba @xcite spin - orbit interaction ( soi ) : in a two - dimensional electron gas confined by an asymmetric potential well , the strength of this soi can be varied by an electric field perpendicular to the plane in which the electrons move @xcite . an early proposal of a spin field - effect transistor @xcite used the rashba soi to control the spin precession of electrons moving in quasi - one - dimensional wires . some of the most striking quantum effects arise due to interference , which is best demonstrated in quantum networks containing loops . indeed , interference due to the rashba soi has been measured on a nanolithographically - defined square loop array @xcite . recently , several theoretical groups proposed spin filters based on a _ single _ loop , subject to both an electric and a magnetic ( aharonov - bohm ( ab ) @xcite ) perpendicular fields ( e.g refs . however , such devices produce a full polarization of the outgoing electrons only for _ special values _ of the two fields . later work considered the effects of the rashba soi on the conductance of chains of loops . these included studies of chains of diamond - like loops @xcite , and studies of finite chains of circular loops @xcite . although both studies showed some destructive interference due to the soi , they did not concentrate on the tuning of the fully polarized spins which can be obtained in certain parameter ranges . recently @xcite , we proposed a spin filter based on a periodic chain of diamond - like square loops , connected to each other at opposite corners [ see fig . [ 1 ] ] . unlike the above earlier papers , which used a continuum description for the wires on the network , we used a simple tight - binding model , with quantum dots ( or ` atoms ' ) only at the nodes of the square diamonds . this allowed us to obtain transparent analytical expressions for the ballistic conductance through the chain and for the outgoing spin polarization . we found that a combination of both the rashba soi and the ab flux through each loop can result in destructive interference , which can block the transmission of all the spin components except one , which is then polarized at a tunable direction . technically , this single spin direction is associated with a single propagating wave solution of the schrdinger equation , while all the other solutions involve evanescent modes .    here we extend our analysis of this diamond - like chain in several directions . first , we replace the previous square loops by rhombi , with a general angle @xmath0 ( fig . [ 1 ] ) . it turns out that the filter is more efficient for @xmath1 . second , we generalize our previous study , by allowing different site energies ( controlled by appropriate gate voltages ) on the various sites in the unit cell ( i.e. sites @xmath2 and @xmath3 in fig . [ 1 ] ) . different site energies at sites @xmath4 and @xmath3 turn out to have drastic effects on the ballistic conductance . third , we propose using this filter at fixed electric and magnetic fields , controlling the outgoing polarization using a gate voltage . ( in ref . @xcite we worked at fixed energy , and varied the magnetic and electric fields . ) fourth , we replace each edge of each rhombus by a tight - binding chain of ` atoms ' ( or quantum dots ) . in this context , we compare our tight - binding approach with the continuous quantum network approach used in earlier work on the same geometry @xcite . as we discuss elsewhere @xcite , the two approaches are not equivalent . section 2 outlines the tight - binding model which we use for solving the schrdinger equation on the periodic chain of generalized diamonds . section 3 presents results for the polarization of the electrons in the regions where they are fully polarized . we compare our tight - binding approach to earlier continuum network models in sec . 4 , and summarize our results in sec . 5 the basic theory was presented in ref . however , for the generalizations introduced here we find it easier to use the ( rotated ) coordinate axes shown in fig . [ 1 ] . setting the opening angle of each diamond to @xmath0 , the coordinates of the sites in the @xmath5th unit cell become @xmath6 and @xmath7 , where @xmath8 is the basic step along the @xmath9axis . we start with the simplest tight - binding model , which has ` atomic ' sites only at the corners of the diamonds ( this will be extended below ) . the hopping @xmath10 unitary matrices within the @xmath5th diamond thus become @xmath11 where @xmath12 denotes the site @xmath13 , @xmath14 is the vector of pauli matrices , @xmath15 ( @xmath16 represents the strength of the rashba term ) , @xmath17 , @xmath18 and @xmath19 is the number of flux units through each diamond . for each bond , the hopping matrix must be multiplied by the hopping energy , @xmath20 ( below we measure all energies in units of @xmath21 ) .    within the @xmath5th diamond , the schrdinger equations for the spinors @xmath22 and @xmath23 are @xmath24 except for the special energies @xmath25 , which represent dispersionless solutions ( not shown in the figures ) , we express @xmath26 and @xmath23 in terms of @xmath27 and @xmath28 , and substitute into the equations for @xmath27 . we end up with effective one - dimensional equations , @xmath29 with @xmath30 , @xmath31 , and with the non - unitary @xmath32 matrix @xmath33,\nonumber\\ & 2b_y=2a^{}_+ c s \cos\beta,\ \ \ 2b_z =- ia^{}_-s^2\sin(2\beta),\label{db}\end{aligned}\ ] ] with @xmath34 , @xmath35 and @xmath36 assuming a propagating wave , @xmath37 , where @xmath38 is a normalized spinor , we find that this spinor must obey @xmath39 , with @xmath40 where @xmath41 , and @xmath42 . therefore , @xmath43 is an eigenstate of @xmath44 , with the unit vector @xmath45 : @xmath46 , and we have @xmath47 . thus , @xmath48 and @xmath49 must obey the equation @xmath50 . at fixed @xmath49 , this is a polynomial of degree 6 in @xmath48 , so that one expects 6 energy bands . in the special symmetric case where @xmath51 , partially treated in ref . @xcite , two of these solutions represent dispersionless solutions at @xmath52 , and thus one is left with only four bands ( whose shape and location varies with @xmath53 ) . the value of @xmath54 only represents a shift in energy , so we fix @xmath55 . for @xmath56 the dispersionless modes become dispersive , and in general the spectrum also becomes asymmetric with respect to @xmath57 and to @xmath58 . an example of this phenomenon is shown in fig . [ spec ] , for the square diamond ( @xmath59 ) .    as explained in @xcite , the ballistic conductance at a given energy @xmath48 is equal to @xmath60 , where @xmath61 is the number of propagating wave solutions which move in one direction . to study @xmath61 , we now fix @xmath48 and solve the spectrum equation for @xmath49 . in the general case , this equation turns into a quartic equation in @xmath62 . this equation reduces to a quadratic equation for @xmath51 . out of the solutions for @xmath49 we count only the right - moving propagating modes , with real @xmath49 and with a positive velocity @xmath63 . as in @xcite , we again find ranges of @xmath48 with @xmath64 or @xmath65 . generally @xmath61 depends on all three parameters @xmath66 and @xmath67 . unlike ref . @xcite , where we fixed @xmath48 and presented results as functions of @xmath68 or @xmath67 , here we fix @xmath67 and fig . [ contour ] presents contour plots of @xmath69 at @xmath70 . as already indicated by fig . [ spec ] , changing @xmath71 can open the large gap which existed in the symmetric case near @xmath72 . this can be seen by comparing the top two plots in fig . [ contour ] : the right hand side plot , for @xmath73 and @xmath74 , is asymmetric with respect to changing the sign of @xmath48 , and it exhibits non - zero ballistic conductance at small positive energy . the gate voltage governing @xmath71 can thus be used efficiently to vary the ballistic conductance between zero and non - zero values .    . four full lines : @xmath75 . six dashed lines : @xmath76 . all energies are in units of @xmath21.,width=226 ] in the regions with @xmath77 we have only one propagating mode . as explained above , each mode is associated with a single spinor @xmath38 . the spinor of an electron coming from the left will in principle be written as a linear combination of all the four ( or six ) solutions . however , when @xmath77 all the right moving modes except one are evanescent , i.e. they decay with distance . therefore , the regions with @xmath77 represent full polarization of the conducting electrons . for the symmetric case , fig . [ contour ] also compares between three values of the rhombus angle @xmath78 . interestingly , the square diamond ( @xmath59 ) is not the optimal filter ; increasing @xmath78 broadens the regions with @xmath77 , where we have full polarization of the electrons . therefore , we present below results for @xmath79 . looking at each panel in fig . [ contour ] , we can identify cuts for which there are broad regions with @xmath77 . for each cut , the spin polarization is given by @xmath80 . figure [ kapspin ] shows these spin components , as a function of @xmath48 at fixed @xmath81 ( the spins are fully aligned in the @xmath82direction for @xmath83 ) and as a function of @xmath68 at fixed @xmath84 . as one can see , small changes in @xmath48 ( determined by the fermi energy ) or in @xmath68 ( determined by the voltage which fixes the strength of the rashba soi ) can cause jumps in @xmath85 between large positive and negative values .     and @xmath86 of the polarized propagating mode and the evanescent inverse decay length ( in units of @xmath87 ) versus @xmath48 at @xmath81 ( left ) and versus @xmath88 at @xmath84 ( right ) . all plots have @xmath79 . data are shown only in regions where @xmath77.,title="fig:",width=136 ]   and @xmath86 of the polarized propagating mode and the evanescent inverse decay length ( in units of @xmath87 ) versus @xmath48 at @xmath81 ( left ) and versus @xmath88 at @xmath84 ( right ) . all plots have @xmath79 . data are shown only in regions where @xmath77.,title="fig:",width=136 ] +   and @xmath86 of the polarized propagating mode and the evanescent inverse decay length ( in units of @xmath87 ) versus @xmath48 at @xmath81 ( left ) and versus @xmath88 at @xmath84 ( right ) . all plots have @xmath79 . data are shown only in regions where @xmath77.,title="fig:",width=136 ]   and @xmath86 of the polarized propagating mode and the evanescent inverse decay length ( in units of @xmath87 ) versus @xmath48 at @xmath81 ( left ) and versus @xmath88 at @xmath84 ( right ) . all plots have @xmath79 . data are shown only in regions where @xmath77.,title="fig:",width=136 ] +   and @xmath86 of the polarized propagating mode and the evanescent inverse decay length ( in units of @xmath87 ) versus @xmath48 at @xmath81 ( left ) and versus @xmath88 at @xmath84 ( right ) . all plots have @xmath79 . data are shown only in regions where @xmath77.,title="fig:",width=136 ]   and @xmath86 of the polarized propagating mode and the evanescent inverse decay length ( in units of @xmath87 ) versus @xmath48 at @xmath81 ( left ) and versus @xmath88 at @xmath84 ( right ) . all plots have @xmath79 . data are shown only in regions where @xmath77.,title="fig:",width=136 ] +    in practical situations , one will not use an infinite chain of diamonds . as discussed in ref . @xcite , the spin polarization is maintained even for a finite chain , provided we use adiabatic contacts at the exit . however , this finite chain must be long enough so that the evanescent modes will decay before the electrons come out . in the symmetric case , @xmath62 is found from solving the quadratic equation @xmath89 . for @xmath77 , one of the solutions has @xmath90 , and therefore a real @xmath49 , while the other solution has @xmath91 , and therefore an imaginary @xmath92 . the bottom panels in fig . [ kapspin ] show the imaginary part of @xmath49 for these other modes , denoted by @xmath93 . as one can see , there are broad regions in which @xmath94 , so that a small number of diamonds suffices for the evanescent modes to decay . interestingly , @xmath93 diverges to infinity as @xmath68 approaches special values , for which the coefficient of @xmath95 in the above quadratic equation approaches zero . writing this equation as @xmath96 , the solutions for small @xmath97 are @xmath98 and @xmath99 . clearly , @xmath93 diverges as @xmath97 approaches zero . using the specific relations which follow eq . ( [ h ] ) , we identify @xmath100 . it is then easy to check that the condition @xmath101 is equivalent to @xmath102 . when this condition is obeyed , @xmath103 and @xmath104 have a vanishing eigenvalue , which means that the corresponding eigenvector is completely blocked [ see eq . ( [ ren ] ) ] . interestingly , ref . @xcite found similar blocking for finite chains at special values of the parameters . we now generalize our model , by replacing each edge of a diamond by @xmath105 bonds in series . consider a single ` large ' bond , e.g. the bond @xmath106 in fig . [ 1 ] . solving the appropriate one - dimensional tight - binding model along this bond , we find that the spinor on the site @xmath107 , @xmath108 [ @xmath109 , with @xmath110 and @xmath111 , is given by @xmath112}{\sin(kam)}[u^{\dagger}_{ab}]^{m / m}\psi^{}_a\nonumber\\ & + \frac{\sin(kam)}{\sin(kam)}[u^{}_{ab}]^{(m - m)/m}\psi^{}_b,\label{psiab}\end{aligned}\ ] ] where @xmath113 is the same as in eq . ( [ eq1 ] ) , @xmath114 is the new lattice constant , and @xmath115 is the wave vector for the one - dimensional solution , related to the energy @xmath48 and to the new elementary hopping energy @xmath21 via @xmath116 . we next discuss a general node on the network , @xmath117 . the tight - binding equation at this node is @xmath118^{1/m}\psi^{(uv)}(1),\label{eq7}\end{aligned}\ ] ] where we assumed that all the ` large ' bonds are equivalent to each other , having the same number @xmath105 of internal bonds , the same lattice constant @xmath97 and the same hopping energy @xmath21 . substituting eq . ( [ psiab ] ) on the right hand side , it is straightforward to rewrite eq . ( [ eq7 ] ) as @xmath119 with @xcite @xmath120\bigr ) , \label{eu}\end{aligned}\ ] ] where @xmath121 is the number of bonds meeting at site @xmath117 . the generalization to arbitrary bonds is obvious . equations ( [ neweq ] ) look exactly like our tight - binding equations ( [ eq2 ] ) , provided we replace @xmath122 by @xmath123 . therefore , we might expect some similarities in the solutions . the equation for @xmath124 in ( [ eq2 ] ) now has @xmath125 on the left hand side , and therefore the dispersionless modes contain all the solutions of @xmath126 . with @xmath127 , this yields @xmath128 . except for these energies , we again eliminate the side site spinors , restricting ourselves to the symmetric case , @xmath51 . in this case , we also have @xmath129 . it is then convenient to separate the common factor @xmath130 from eq . ( [ db ] ) , and rewrite the generalized eq . ( [ ren ] ) in the form @xmath131 with @xmath132 , and with @xmath133 , where now @xmath134 . the symmetric cases which we described above are characterized by the same matrix @xmath135 . since all the spin physics described above resulted only from the matrix @xmath135 , which does not depend on the energy @xmath48 , all of that discussion will remain unchanged . the only effect of adding the internal bonds on each ` large ' bond appears in the new parameter @xmath136 , which is now given by @xmath137 as before , the spectrum is determined by @xmath138 where @xmath139 and @xmath140 . for each value of @xmath49 , ( [ newspec ] ) determines two values for @xmath136 , which are independent of @xmath105 and of @xmath48 . we then solve each of these equations for all possible values of @xmath141 , and obtain the energies of the various bands via @xmath116 . bonds on each ` large ' bond , with @xmath142 . top : @xmath143 ( left ) , @xmath144 ( right ) . bottom : @xmath145 . all energies are in units of @xmath21.,title="fig:",width=136 ]   bonds on each ` large ' bond , with @xmath142 . top : @xmath143 ( left ) , @xmath144 ( right ) . bottom : @xmath145 . all energies are in units of @xmath21.,title="fig:",width=136 ] +   bonds on each ` large ' bond , with @xmath142 . top : @xmath143 ( left ) , @xmath144 ( right ) . bottom : @xmath145 . all energies are in units of @xmath21.,title="fig:",width=136 ]   bonds on each ` large ' bond , with @xmath142 . top : @xmath143 ( left ) , @xmath144 ( right ) . bottom : @xmath145 . all energies are in units of @xmath21.,title="fig:",width=136 ]    for the special case @xmath146 , eq . ( [ lam ] ) reduces to @xmath147 . examples of the spectra for this case and for several values of @xmath105 are shown in fig . [ newe ] . clearly , the number of bands increases with @xmath105 . however , the basic qualitative shapes of @xmath148 for all the bands are similar to each other . thus , we expect that within each band we would reproduce the filtering properties described in sec . however , the energy scales for each band become narrower as @xmath105 increases . this narrowing is most pronounced for the lowest band , which always appears below the band of the one - dimensional solutions , @xmath149 . within our tight - binding model , this narrow band involves an imaginary value of @xmath115 , @xmath150 , implying very small wave functions in the middle of each ` large ' bond . this imaginary solution results from the solution of @xmath151 . at large @xmath105 , @xmath136 becomes negligible , and the two bands converge to a single band , with @xmath152 and thus @xmath153 . the bottom panels in fig . [ newe ] show the spectrum for @xmath145 . the right hand side panel zooms on the two lowest bands , demonstrating that these bands maintain their qualitative features even as @xmath105 increases . the lowest band does indeed narrow down , becoming dispersionless in the limit @xmath154 , where the equation @xmath155 yields the solution @xmath156 , with @xmath157 . in contrast to our calculations , refs . @xcite and @xcite used a continuum wire model for each ` large ' bond on each diamond . their solution for the spinor at distance @xmath158 from node @xmath97 on the bond @xmath106 is given by @xmath159}{\sin(kl)}[u^{\dagger}_{ab}]^{x / l}\psi^{}_a\nonumber\\ & + \frac{\sin(kx)}{\sin(kl)}[u^{}_{ab}]^{(l - x)/l}\psi^{}_b.\label{psiabb}\end{aligned}\ ] ] since the electron on each bond is now free , its energy is given by @xmath160 , and bercioux _ et al . _ plot @xmath161 . as usual with tight - binding equations , this solution coincides with our tight - binding solution ( [ psiab ] ) in the limit @xmath162 , keeping @xmath163 and @xmath164 . having found these solutions , refs . @xcite and @xcite proceed to use the neumann boundary conditions at the nodes : @xmath165 with these conditions , they end up with equations like our ( [ neweq ] ) , but with our @xmath123 replaced by @xmath166 . thus , they would have @xmath167 , and therefore @xmath168 . these results , with @xmath169 , are reproduced in the left panel of fig . again , the qualitative shape of each band looks similar to ours . however , this approach can not reproduce the lowest band with imaginary @xmath115 which we found for finite @xmath105 .    , right ) , with @xmath142 and with energies in units of @xmath21.,title="fig:",width=136 ]  , right ) , with @xmath142 and with energies in units of @xmath21.,title="fig:",width=136 ] +    as discuss in ref . @xcite , there is no way to make the tight - binding model and the continuum models identical for any finite @xmath105 . however , in the limit @xmath162 one can modify the tight - binding model , by setting a site energy @xmath170 . with this special value , our eq . ( [ lam ] ) becomes @xmath171 , which becomes identical to the expression used in refs . @xcite and @xcite when @xmath172 . unfortunately , we know of no good reason to choose this particular value for the site energy in the tight - binding model . the right hand panel in fig . [ berc ] shows results of the modified tight - binding model for @xmath145 . except for the dispersionless band below the continuum , we expect these results to approach those in the left hand side panel as @xmath162 . reference @xcite already compared the results of a chain of diamonds to a single diamond . as also noted in ref . @xcite , having more loops in series broadens the parameter regions which yield full polarization . reference @xcite also discussed the conditions for having full filtering on a chain of finite length . here we have extended that discussion by showing that there exist broad regions with relatively short evanescent decay lengths , so that one can obtain filtering with relatively short chains . in addition , we have shown that the filtering results are quite robust : in addition to the parameters discussed in ref . @xcite , the filtering persists upon changing many additional parameters ( e.g. the opening angle of each diamond ) . we also mention the asymmetry of the spectra in the non - symmetric case , which implies that a large ( positive or negative ) bias voltage between the left and right hand ends of the device can yield different currents in the two directions .
we present several new results , extending our recent proposal of a spin filter based on a tight - binding model for a periodic chain of diamond - like loops [ phys . 
 rev . 
 b * 78 * , 125328 ( 2008 ) ] . in this filter , 
 the rashba spin - orbit interaction ( which can be tuned by a perpendicular gate voltage ) and the aharonov - bohm flux ( due to a perpendicular magnetic field ) combine to select only one propagating ballistic mode . for this mode , 
 the electronic spins are fully polarized along a direction that can be controlled by the electric and magnetic fields and by the electron energy . 
 all the other modes are evanescent . 
 generalizing the square diamonds into rhombi with arbitrary opening angles , we find that increasing these angles widens the parameter range for efficient filtering . a different gate voltage on the two sides of each rhombus is found to delocalize the electrons for energies on one side of the band center . 
 we also compare our tight - binding model with models which use continuous quantum networks of one - dimensional wires , and find coincidence only when one chooses particular site energies at the nodes of the network .