LETTERS W=20v3a 4=πr/40 0.5 0.5 1.0 1.5 阳/r Figure 3 Dispersion relation of a graphene strip with zigzag edges.The spacing of the low--lying modes approaches△=(i/2)√3πta/Wfor W/a>1.The zeroth and first modes have a larger spacing,approaching 3A/2 for W/a1.The Figure 2 Schematic diagram of the valley valve(top)and corresponding vertical lines mark the valley centres at k=2/3a and 4/3a. potential profile (bottom).The current through the constriction is blocked if =E-U and uR=E-U have opposite signs. zeroth mode,n=0,lies in a single valley fixed by the direction of propagation.The conductance of the constriction is determined by conduction bands in the wide regions and in the valence band the Landauer formula inside the constriction.Two valley filters in series,one acting as a polarizer and the other as an analyser,can block the current if they have the opposite polarity(see Fig.2),demonstrating that a QPC Gs2e、 =一N can operate as a 'valley valve'-a purely electronic counterpart of m=-N the magneto-electronic spin valve.This extends to a 2D geometry The valley polarization of the transmitted current is quantified by the findings in a 1D geometry by Wakabayashi and Aokil2.We emphasize that their earlier work could not have demonstrated the P=+(-T selective population of a single valley-simply because valleys do not exist independently in 1D. ∑-N工. Our calculations start from the tight-binding model of where we consider the case (illustrated in Fig.1)that the zeroth graphene,with hamiltonian mode lies in the first valley.The polarization PE[-1,1],with P=1 if the transmitted current lies fully in the first valley and P=-1 if H=〉闭+》∑U(l. it lies fully in the second valley. We have calculated the transmission matrix numerically by adapting to the honeycomb lattice the method developed by Ando The hopping matrix element t=-t if the orbitals li)and li)are for a square lattice7.The results are shown in Figs 4 and 5.We have nearest neighbours on the honeycomb lattice,otherwise =0.The fixed the width of the wide regions at Wo=70v3a (in units of electrostatic potential energy U:=U(x)varies only along the axis the lattice spacing a).The electrochemical potential in the wide of the constriction.It equals U in the wide regions and rises to regions is set at Er-U==/3,corresponding to 2N+1=29 Uo inside the constriction.We smooth the stepwise increase of the propagating modes.The narrow region has width W=20v3a.We potential over a length L.,according to the function measure the electrochemical potential E-Uo=uo in the narrow region in units of the mode spacing if x<-Ls/2, 1 .(x) sin(x/L)if x<L/2, A=jV3xta/W=xhv/W ifx>L/2. (with v=(1/2)3ra/h =3 x 106 ms-being the energy- The potential barrier U(x)=Ue+(Uo-U)[eL,(x)-eL independent velocity in graphene).For our parameters A=/40, (x-L)]is rectangular for L =0(solid line in Fig.1,bottom panel), as indicated in Fig.3. whereas it has a sinus shape for L =L(dashed line). The operation of the valley filter is demonstrated in Fig.4.The The dispersion relation of the honeycomb lattice in a strip top panel shows the conductance,whereas the bottom panel shows with zigzag edges is shown schematically in Fig.1(top panel)and the valley polarization-both as a function of the electrochemical exactly in Fig.3.The wide regions support 2N+1 propagating potential,uo,in the narrow region.For positive uo,the current modes at the Fermi energy,Er,which form a basis for the flows entirely within the conduction band,and we obtain plateaus transmission matrix,t.Modes n=1,2,....N lie in the first of quantized conductance at odd multiples of 2e2/h(as predicted valley (with longitudinal wavevector kae(t,2t)),whereas modes by Peres et al3).Smoothing of the potential step improves the n=-1,-2,...,-N lie in the second valley (with ka(0,)).The flatness of the plateaus(compare the solid and dashed lines).The nature physics I VOL 3|MARCH 2007 I www.nature.com/naturephysics 173 @2007 Nature Publishing GroupLETTERS + – x x UL UR L1 L2 U EF Figure 2 Schematic diagram of the valley valve (top) and corresponding potential profile (bottom). The current through the constriction is blocked if μL = EF −UL and μR = EF −UR have opposite signs. conduction bands in the wide regions and in the valence band inside the constriction. Two valley filters in series, one acting as a polarizer and the other as an analyser, can block the current if they have the opposite polarity (see Fig. 2), demonstrating that a QPC can operate as a ‘valley valve’—a purely electronic counterpart of the magneto-electronic spin valve. This extends to a 2D geometry the findings in a 1D geometry by Wakabayashi and Aoki12. We emphasize that their earlier work could not have demonstrated the selective population of a single valley—simply because valleys do not exist independently in 1D. Our calculations start from the tight-binding model of graphene, with hamiltonian H = i,j τij|ij|+ i Ui|ii|. The hopping matrix element τij = −τ if the orbitals |i and |j are nearest neighbours on the honeycomb lattice, otherwise τij =0. The electrostatic potential energy Ui = U(xi) varies only along the axis of the constriction. It equals U∞ in the wide regions and rises to U0 inside the constriction. We smooth the stepwise increase of the potential over a length Ls, according to the function ΘLs(x) = ⎧ ⎨ ⎩ 0 if x < −Ls/2, 1 2 + 1 2 sin(πx/Ls) if |x| < Ls/2, 1 if x > Ls/2. The potential barrier U(x) = U∞ + (U0 − U∞)[ΘLs(x) − ΘLs (x−L)] is rectangular for Ls =0 (solid line in Fig. 1, bottom panel), whereas it has a sinus shape for Ls = L (dashed line). The dispersion relation of the honeycomb lattice in a strip with zigzag edges is shown schematically in Fig. 1 (top panel) and exactly in Fig. 3. The wide regions support 2N + 1 propagating modes at the Fermi energy, EF, which form a basis for the transmission matrix, t. Modes n = 1, 2,..., N lie in the first valley (with longitudinal wavevector ka ∈ (π,2π)), whereas modes n = −1,−2,...,−N lie in the second valley (with ka ∈ (0,π)). The 0 0 0.5 0.5 1.0 1.5 ka/π E/ Δ 3Δ W = 20 3a = Δ π τ /40 τ Figure 3 Dispersion relation of a graphene strip with zigzag edges. The spacing of the low-lying modes approaches Δ ≡ (1/2)√ 3 πτa/W for W/a  1. The zeroth and first modes have a larger spacing, approaching 3Δ/2 for W/a 1. The vertical lines mark the valley centres at k = 2π/3a and 4π/3a. zeroth mode, n = 0, lies in a single valley fixed by the direction of propagation. The conductance of the constriction is determined by the Landauer formula G = 2e2 h N n=−N Tn , Tn = N m=−N |tnm| 2 . The valley polarization of the transmitted current is quantified by P = T0 +N n=1 (Tn −T−n ) N n=−N Tn , where we consider the case (illustrated in Fig. 1) that the zeroth mode lies in the first valley. The polarization P ∈ [−1,1], with P =1 if the transmitted current lies fully in the first valley and P = −1 if it lies fully in the second valley. We have calculated the transmission matrix numerically by adapting to the honeycomb lattice the method developed by Ando for a square lattice17. The results are shown in Figs 4 and 5. We have fixed the width of the wide regions at W∞ = 70√3 a (in units of the lattice spacing a). The electrochemical potential in the wide regions is set at EF−U∞ ≡μ∞ =τ/3, corresponding to 2N +1=29 propagating modes. The narrow region has width W =20√3 a. We measure the electrochemical potential EF −U0 ≡ μ0 in the narrow region in units of the mode spacing Δ ≡ 1 2 √ 3 πτa/W = πhv¯ /W (with v = (1/2) √3τa/h¯ = 3 × 106 m s−1 being the energy￾independent velocity in graphene). For our parameters Δ=πτ/40, as indicated in Fig. 3. The operation of the valley filter is demonstrated in Fig. 4. The top panel shows the conductance, whereas the bottom panel shows the valley polarization—both as a function of the electrochemical potential, μ0, in the narrow region. For positive μ0, the current flows entirely within the conduction band, and we obtain plateaus of quantized conductance at odd multiples of 2e2 /h (as predicted by Peres et al.13). Smoothing of the potential step improves the flatness of the plateaus (compare the solid and dashed lines). The nature physics VOL 3 MARCH 2007 www.nature.com/naturephysics 173 Untitled-1 2 13/2/07, 12:35:51 pm