Since the unidirectional motion of the edge states prevents reflections, the transmission matrixt from I1to I2is the product of the transmission matricest1from I1 to NS and
t2from NS to I2. Each of the matricestp is a22unitary matrix, diagonal in the basis j ˙pi:
tpDei pj CpihCpj Cei
0
pj
pih pj: (3.19)
The phase shiftsp; p0 need not be determined. Usingjh1j ˙2ij2D 12.1˙12/, we obtain fromt Dt2t1the required transmission probabilities
The D1 TeeD 21.1 12/: (3.20)
Substitution into Eq. (3.18) gives our central result (3.1).
Referring to Fig. 3.1, we see thatGNS D 0 in the case (a) of a superconducting
contact to a single edge (1 D2) — regardless of whether the edge is zigzag or arm- chair. In the case (c) of a contact between a zigzag and an armchair edge we have
12 D 0) GNS D 2e2= h. The case (b) of a contact between two opposite edges
has1 D 2 ) GNS D 4e2= h if both edges are zigzag; the same holds if both
edges are armchair separated by a multiple of three hexagons (as in the figure); if the number of hexagons separating the two armchair edges is not a multiple of three, then
12D1=2)GNSDe2= h.
Intervalley relaxation at a ratetends to equalize the populations of the two degener-
ate modes propagating along the NS interface. This becomes appreciable ifL=v0&1, withLthe length of the NS interface andv0 D „ 1d "0=dq ' min.v=2;
p
2 lm=„/ the velocity along the interface. The density matrixD0.1 e L=v0/C1e L=v0 then contains a valley isotropic part0 /0withTee DTeh D1=2and a nonequilib- rium part0/ j1ih1jwithTee; Tehgiven by Eq. (3.20). The conductance then takes the form GNS D 2e 2 h 1 e L=v0cos‚ : (3.21)
A nonzero conductance when the supercurrent covers a single edge (‚D 0) is thus a
direct measure of the intervalley relaxation.
3.4
Conclusion
In conclusion, we have shown that the valley structure of quantum Hall edge states in graphene, which remains hidden in the Hall conductance, can be extracted from the current that flows through a superconducting contact. Since such contacts have now been
3.4 Conclusion 49 fabricated succesfully [47, 48], we expect that this method to detect valley polarization can be tested in the near future.
Chapter 4
Theory of the valley-valve ecect in graphene
nanoribbons
4.1
Introduction
The massless conduction electrons in a two-dimensional carbon lattice respond differ- ently to an electric field than ordinary massive electrons do. Because the magnitudev
of the velocity of a massless particle is independent of its energy, a massless electron moving along the field lines cannot be backscattered — since that would requirevD0
at the turning point. The absence of backscattering was discovered in carbon nanotubes [49], where it is responsible for the high conductivity in the presence of disorder.
A graphene nanoribbon is essentially a carbon nanotube that is cut open along the axis and flattened. One distinguishes armchair and zigzag nanotubes, depending on whether the cut runs parallel or perpendicular to the carbon-carbon bonds. The edges of the nanoribbon fundamentally modify the ability of an electric field to backscatter elec- trons. As discovered in computer simulations by Wakabayashi and Aoki [50], a potential step in a zigzag nanoribbon blocks the current when it crosses the Fermi level, forming a
p-njunction (= a junction of states in conduction and valence band). The current block- ing was interpreted in Ref. [51] by analogy with the spin-valve effect in ferromagnetic junctions [52]. In this analogy the valley polarization in a zigzag nanoribbon plays the role of the spin polarization in a ferromagnet — hence the name “valley-valve” effect.
It is the purpose of this chapter to present a theory for this unusual phenomenon. A theory is urgently needed, because the analogy between spin valve and valley valve fails dramatically to explain the computer simulations of Fig. 4.1: The current blocking by the p-njunction turns out to depend on the parity of the numberN of atom rows
in the ribbon. The current is blocked whenN is even (zigzag configuration), while it
is not blocked whenN is odd (anti-zigzag configuration, see Fig. 4.2). This even-odd
difference (first noticed in connection with the quantum Hall effect [53]) is puzzling since zigzag and anti-zigzag nanoribbons are indistinguishable at the level of the Dirac
52 Chapter 4. Theory of the valley-valve effect
Figure 4.1: ConductanceG of a zigzag nanoribbon containing a potential stepU D 1
2U0Œ1Ctanh.2x=d /. The red and blue curves are obtained by computer simulation of the tight-binding model of graphene, with parametersd D 10 a,EF D 0:056 t, where
ais the lattice constant and t is the nearest-neighbor hopping energy. Upon varying U0the conductance switches abruptly to zero when the Fermi levelEF is crossed and ap-njunction is formed (red solid curve; the deviation from an ideally quantized step function is.10 7). This “valley-valve” effect occurs only for an even numberN of car-
bon atom rows (zigzag configuration). WhenN is odd (anti-zigzag configuration), the
conductance remains fixed at2e2= h(blue dotted curve, again quantized within10 7). equation1 [21], which is the wave equation that governs the low-energy dynamics in
graphene.
1The dependence of boundary conditions on the numberNof atoms across the ribbon is a key distinction
between zigzag and armchair edges. The boundary condition of the Dirac equation for an armchair nanoribbon depends onN(modulo 3), but there is noN-dependence for a zigzag nanoribbon.