-50 -40 -30 -20 -10 0 TMR (%) N=15 -10 -8 -6 -4 -2 0 TMR (%) N=21 -5 0 5 10 15 20 d (Å) -2 -1.5 -1 -0.5 0 TMR (%) N=27 6 FIG. 7: Tunneling magnetoresistance as a function of the lon- gitudinal distance between H atoms for three different arm- chair graphene nanoribbon widths. The H atoms are located on opposite edges of the ribbon. on the opposite edges of an L = 57 . 0 ˚ A AGNR for three different widths. Large (and negative) values are also obtained for the tail-to-tail configurations in the range of distances explored. As can be appreciated, the TMR logically decreases with the ribbon width. A critical assesment of the practical consequences of the results presented is due at this point: zipped metallic c arbon nanotubes $^{31}$, as suggested in Ref. $^{32}$. A th ird possibility consists in gating selectively a sem iconducting AGNR as previously done for nanotub es $^{33}$. IV. PRACTICAL CONSIDERATIONS V. SUMMARY In summary, we have shown, as a proof of princi- ple, that hydrogenated AGNR’s at small concentrations can exhibit magnetoresistive properties without invok- ing purely edge-related physics. Our results, which are deeply rooted in Lieb’s theorem, provide further evidence for the wide applicability of this theorem beyond the bipartite lattice Hubbard model for which was demon- strated. We have also shown that hydrogenation near the edge presents advantages with respect to bulk hydro- genation both from energetic and electronic standpoints. This aspect, especific to nanoribbons, might favor the use • First, the TMR results presented above have been obtained for a particular type of AGNR het- erostructure and rely on the existence of metal- lic AGNR’s. Calculations where the relaxation of the atomic structure on the edges is considered re- veal that a gap always opens even for the nom- inally metallic armchair nanoribbons $^{30}$, compro- mising the applicability of these nanoribbons as electrodes. Graphene-based metallic leads, how- ever, can be found in recent literature. For in- stance, zigzag graphene nanoribbons with passi- vated edges are metallic $^{17}$. The physics described in this work does not rely on the edge termination of the nanoribbons and could be reproduced in fully passivated zigzag nanoribbons as well. Another alternative can be based on using partially un- • The third caveat relates to the stability of the atomic configurations. Given the tendency for H atoms to form lowest-energy non-magnetic aggregates $^{28}$, the existence of magnetically active dilute ensembles of adsorbed H atoms requires cer- tain conditions. For instance, working at reason- able low temperatures prevents H atoms from di- fusing after adsorption $^{34}$. Another possibility is one intrinsic to AGNR’s: The binding energy is larger for H atoms close to the edge than in the bulk of the nanoribbon. Related to this is the fact that the mo- bility of H decreases significantly near the edge, re- ducing the possiblity of formation of lowest-energy non-magnetic aggregates near the edge $^{28}$. As we have shown, it is precisely the hydrogenation near the edge that favors the appearance of sizable MR and thus an actual experimental verification. • From the LSDA calculations we note that the en- ergy difference b etween the F and AF states, or exchange couplin g energy, can be as large as tens of meV for short d istances, particularly for head-to- head configuratio ns. However, as far as magnetore- sistive properties is concerned, exchange coupling energies above 1 meV are of no practical use since the Zeeman energ y gain per spin in a magnetic field B is 0 . 067 gµ$_{B}$B m eV T − 1 and fields higher than 10 T are hardly acce sible in the lab. The AF-F transi- tion is therefore p ractically forbidden below d ≈ 65 ˚ A and d ≈ 40 ˚ A in the examples shown in Fig. 3(a) and (b), respectiv ely. At larger distances the fields necessary to indu ce the AF-F transition can be as small as needed, but also is the associated MR as shown in Fig. 6(a ) and (c). This is not a problem, however, when th e H atoms are coupled tail-to-tail as, e.g., in the ca ses where they are located on op- posite edges. The re, as shown in Fig. 7, a TMR as high as ≈ 50% ca n be obtained at reasonably small exchange couplin g energies [see inset in Fig. 3(c)].