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tion, here the electronic structure will be indirectly af- fected by the field even if its direct influence on the or- bital wave function is neglected. This change reflects in the spin-resolved conductance as shown in Fig. 5(a-b) for d = 32 ˚ A (the dashed line represents the conduc- tance of the clean AGNR). The different total transmis- sion for the F and AF solutions, resulting from adding the two spin channels, results in a positive magnetoresis- tance (MR), MR = G$_{F}$ − G$_{AF}$/G$_{F}$ + G$_{AF}$ , at energies near the bottom and top of the conduction and valence bands, respectively [see Fig. 5(c)]. Similar results are obtained (not shown) for different intermediate distances between H atoms. The right panels in Fig. 5 show the results obtained from the one-orbital mean-field Hubbard model for U = t = 3 eV. Apart from the recovery of the particle-hole symmetry, the results are remarkably simi- lar, validating the use of the latter model for transport calculations in hydrogenated graphene. 

One should note, however, that since the chemical po- tential lies in the middle of the gap, the energy ranges at which MR could manifest itself are not relevant in linear response transport for infinite AGNR’s. A finite bias cal- culation may reveal the MR obtained at those energies, but this is a non-trivial task beyond the scope of this work. The application of a gate voltage is not a prac- tical alternative either since it implies a deviation from charge neutrality which would fill up or empty the local- ized states hosting the unpaired spins and kill the mag- netization. Instead, we propose to explore the possibility of MR at zero bias by considering finite AGNR’s con- nected at the ends to conductive graphene. This is done in our calculations by considering a metallic AGNR with a narrower section in the middle of appropriate width 

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[see Fig. 1(d)]. (Note that in the one-orbital mean-field Hubbard approximation AGNR’s of width N = 3 m − 1, where m is an integer, are metallic, being semiconduct- ing otherwise). In what follows and in the light of the previous results, we restrict ourselves to the one-orbital mean-field Hubbard model. 

In our proposed AGNR heterostructure the difference in the zero-bias tunnel conductance between the F and AF states is now responsible for the appearance of tun- neling MR (TMR), as shown in Fig. 6. Notice that unlike conventional TMR, where the magnetic elements are in the electrodes, in our proposal magnetism is in the barrier. Panels (a) and (b) in Fig. 6 correspond to H atoms placed in the middle of a semiconducting AGNR of length L = 73 . 7 ˚ A and width N = 9 and N = 15, respectively. Both cases refer to head-to-head configura- tions. The obtained TMR changes sign with d , but it is always negligibly small. On the contrary, when placed near the edge [Fig. 6(c)], the TMR is positive and reaches much larger values. This result is for an N = 9, L = 57 . 0 ˚ A AGNR. When the H atoms are now placed in a tail- to-tail configuration near the edge for the same AGNR, the TMR becomes negative and reaches values of up to 100% [see Fig. 6(d)]. As a final example we show in Fig. 7 the TMR for the case where the H atoms are placed 

