Diluted Graphene Antiferromagnet
We study RKKY interactions between local magnetic moments for both doped and undoped graphene. We find in both cases that the interactions are primarily ferromagnetic for moments on the same sublattice, and antiferromagnetic for moments on opposite sublattices. This suggests that at sufficiently low temperatures dilute magnetic moments embedded in graphene can order into a state analogous to that of a dilute antiferromagnet. We find that in the undoped case one expects no net magnetic moment, and demonstrate numerically that this effect generalizes to ribbons where the magnetic response is strongest at the edge, suggesting the possibility of an unusual spin-transfer device. For doped graphene we find that moments at definite lattice sites interact over longer distances than those placed in interstitial sites of the lattice ( vs. ) because the former support a Kohn anomaly that is suppressed in the latter due to the absence of backscattering.
Introduction– Graphene, a two-dimensional honeycomb network of carbon atoms, has recently become a subject of intense interest. The development of practical fabrication techniques for single graphene sheets K.S.Novoselov et al. (2004) has allowed experimental study of this system, confirming its two-dimensional Dirac spectrum in quantum Hall studies and revealing many unique properties K.S.Novoselov et al. (2005); Y.Zhang et al. (2005). In this work we study response functions of graphene in the non-interacting limit, focusing on its consequences for magnetic moments which may be embedded in the system (RKKY interactions). A number of studies of magnetic moments in graphene have identified a tendency toward antiferromagnetic M.Fujita et al. (1996); Son et al. (2006); Pisani et al. (2007); Yazyev and Helm (2007) or ferromagnetic Vozmediano et al. (2005) correlations. These correlations are usually attributed to exchange interactions or other many-body effects. In what follows, we demonstrate that such effects arise even for non-interacting electrons in graphene, and that they are a result of the chirality of the electron states for doped graphene T.Ando (2005), and of the vanishing density of states at the Fermi energy for undoped systems. A non-interacting model of graphene may be justified by its small, density-independent effective parameter, as well as studies suggesting that Fermi liquid theory should work well in physically realizable situations Sarma et al. (2007); Polini et al. (); Gonzalez et al. (1994); Dahal et al. ().
Exchange coupling between local magnetic moments and conduction electrons in metals leads to an effective (RKKY) coupling C.Kittel (1968) among the local moments which oscillates with distance with wavevector ( Fermi wavevector), and an amplitude that decays as in two dimensions, with the separation between impurities. For doped graphene, we shall demonstrate similar behavior, with an important qualitative difference: the sign of the interaction depends on whether the two local moments couple to the honeycomb network on sites of the same sublattice or different ones, and when summed over both sublattices at a fixed distance, the contribution to the RKKY coupling is cancelled, leaving behind a residue that falls off as . Interestingly, analogous studies of the linear response to perturbations that do not distinguish between A and B sublattice sites also result in a behavior Vozmediano et al. (2005); Cheianov and Falko (2006); Wunsch et al. (2006). We will show that the behavior – and the absence of behavior in density response functions – is a direct result of the chiral nature of electrons in graphene.
For undoped graphene () we find the RKKY coupling behaves as at large distances, again with equal magnitudes, that are ferromagnetic when the impurities are on the same sublattice, and antiferromagnetic when on opposite sublattices. This behavior is also connected to that of the full density response, and reflects the vanishing density of states of graphene at the Fermi points. This behavior also dominates in doped graphene for distances where the coupling is greatest in magnitude. Because of this we expect at zero temperature the system will tend to order, with moments oriented in opposite directions for the two sublattices. The state is thus analogous to an ordered state of a dilute antiferromagnet. Analogous behavior has been noted in zigzag graphene ribbons Son et al. (2006) with equal and opposite spin accumulating near the edges, on opposite sublattices. While this effect has been attributed to complicated exchange interactions Son et al. (2006), we present results of simple tight-binding calculations demonstrating that this physics occurs even without interactions, and is a consequence of the unusual nature of the single particle states in graphene.
Hamiltonian, Wavefunction, and RKKY Interaction – The simplest description of graphene is a tight-binding model representing electrons in orbitals of the carbon atoms, which can hop with matrix element between nearest neighbor sites, which are always on opposite sublattices for the honeycomb lattice. The energy states of such a model may be straightforwardly computed T.Ando (2005), and one finds that the spectrum possesses particle-hole symmetry, with a zero energy surface consisting of six points at corners of the Brillouin zone, only 2 of which are inequivalent due to symmetry. When undoped the Fermi surface of graphene passes through these points, which are denoted by and . At long wavelengths, the wavefunctions near each of these points can be described by two component spinor envelope functions , the entries of which are proportional to the amplitude for the electron to be present at unit cell located at on sublattice or . The wavefunctions may be regarded as possessing a quantum number denoting which Dirac point they reside near. The Hamiltonian near such a point is approximately
with the upper (lower) sign denoting the Hamiltonian for states near the () point, and . These Hamiltonians have eigenenergies , and associated eigenstates , where again the upper (lower) sign denotes the solution for the () valley, , and .
Consider local spin degrees of freedom and weakly coupled to electrons in graphene by an exchange interaction at positions at or near sites in sublattices and . In perturbation theory Fischer and Klein (1975); Beal-Monod (1987) the induced interaction between the spins has the form , where , and is the Fourier transform of
Here is the degeneracy due to the valley index, is the number of unit cells in the system, is the Fermi function, and is a factor arising from the matrix element of the spinors associated with the single particle states, which in general depend on the angles and com (a).
Site-Symmetric Moments – When the local moments are located at the centers of the hexagons in the honeycomb network, it becomes appropriate to replace Eq. 1 with a sum, , where is the angle formed by the vectors and . The resulting is then identical to the standard density-density response function, which may be computed straightforwardly Wunsch et al. (2006); Ando (2006); Hwang and Sarma (), with a result that may be expressed conveniently in the form with the chemical potential (assumed positive), , and
Several comments are in order. (1) In spite of the presence of step functions in this expression, its first derivative with respect to is continuous at , in sharp contrast with the situation for a normal two dimensional electron gas (2DEG). The discontinuity in the 2DEG arises from a singularity in the integrand in Eq. 1 (with for a 2DEG) when and – the Kohn anomaly W.Kohn (1959). For graphene, vanishes precisely where the singularity would otherwise occur, removing the discontinuity in the slope. This behavior is a direct result of the chirality of electrons in graphene and the resulting absence of backscattering that it entails T.Ando (2005). (2) For undoped graphene the response vanishes at . We can understand this as follows. The response may be understood as arising from a shift in the chemical potential, plus more generally a part coming from changes in the single particle wavefunctions. However, the total charge of the system cannot shift due to changes in the single particle wavefunctions, in accordance with the Friedel sum rule G.D.Mahan (2000). Moreover, in undoped graphene the response from a differential chemical potential shift vanishes because the density of states at the Fermi energy is zero. Thus there can be no net response. (3) The vanishing of at means that the total population of either spin flavor cannot be changed by a perturbation in undoped graphene, even if the perturbation is different for the two spin directions – as would be the case for a (possibly inhomogeneous) Zeeman coupling. This result is consistent with the observation that graphene ribbons can have an inhomogeneous spin configuration but net spin zero Son et al. (2006).
Site-Specific Moments – Local moments can in many circumstances be more strongly coupled to a specific site in the honeycomb network, which lies on a definite sublattice. One can also consider situations in which the moment is a substitutional impurity, or is an induced moment due to a vacancy in the lattice Vozmediano et al. (2005); Yazyev and Helm (2007). In such cases the coupling among moments has the form , and is given by Eq. 1 with for impurities on the same sublattice, and for impurities on opposite sublattices. We first consider the case of impurities on the same sublattice. Decomposing the response function as the first term, corresponding to undoped graphene, may be shown to have the form
where is the momentum cutoff. The contribution due to doping may also be evaluated, and has the form
In Eq. 4 the derivative is discontinuous at : the chiral overlap factor does not vanish in this case, and one obtains a Kohn anomaly analogous to that of the standard 2DEG. This has important consequences for RKKY coupling in real space, which is proportional to the Fourier transform of Eqs. 3 and 4. For the first of these we find
so that in undoped graphene, moments are ferromagnetically coupled when they are on the same sublattice. The correction due to doping, can be computed in the asymptotic limit (), with the result
A comparison with numerical integration shows that this asymptotic expression works quite well for . The oscillating term proportional to is present because the Kohn anomaly is not suppressed in the relevant response function. A similar behavior was found recently for Friedel oscillations, where the way in which the perturbation breaks the lattice symmetry determines whether they fall off as or Cheianov and Falko (2006). While this behavior is similar to that of the standard 2DEG, it nevertheless differs from the 2DEG in having a density dependent amplitude Beal-Monod (1987).
For moments on opposite sublattices, we can easily compute the coupling by noting that . It immediately follows that
We thus see that the tendency towards ferromagnetic coupling for moments within a distance for impurities on the same sublattice translates into an antiferromagnetic coupling for impurities on opposite sublattices com (b). Moreover because the coupling is strongest for short distances, we expect this to result in a tendency towards antiferromagnetic order at low temperatures when the moment density satisfies . The low temperature state is analogous to that of a dilute antiferromagnet since the moment locations are random in such models. A special feature of the graphene system, however, is that the coupling among the moments can be manipulated via the electron density, which in turn may be controlled by a gate K.S.Novoselov et al. (2004). In particular, added electrons shorten the distance over which the RKKY coupling has a well-defined (i.e., non-oscillating) sign, so that the antiferromagnetic order may be suppressed via doping. It is interesting to note that analogous, albeit simpler, behavior (e.g., ferromagnetic rather than antiferromagnetic ordering) is believed to occur in dilute magnetic semiconductors Brey_2003; Priour and Sarma (2006). The physics associated with the chirality of the single-particle states, as well as the vanishing density of states at the Fermi energy when undoped, give graphene a richer phenomenology.
Numerical Investigations – To test these results we have performed numerical tight-binding calculations on graphene ribbons. We first consider a ribbon with zigzag edges, with a Zeeman coupling () along a line of sites all on one sublattice (A) near the center of the ribbon. This type of perturbation models a line of frozen spins. Figure 1 shows the results for the induced spin density, with A sites shown in red and B sites in black. The main panel is essentially identical for both the doped and undoped cases. In the doped case one can see oscillations of wavevector falling off slowly with distance, which are out-of-phase for the two sublattices. Moreover, the total induced spin vanishes for the undoped case. These properties are in precise agreement with our expectation that summing over sublattices leads to a cancellation of the RKKY oscillations due to the absence of backscattering in graphene, and a vanishing net response as due to the vanishing density of states for undoped graphene.
We also find an interesting result when the perturbation is applied at one of the edges (Fig. 2). Applying a Zeeman field at a single zigzag edge in undoped graphene induces spin in both edges, but in such a way that there is no induced total spin for undoped graphene. This is interesting because the spin state is communicated across the width of the sample even though there is no spin polarization in the bulk. Thus the tendency for undoped graphene to compensate an induced local spin due to a local Zeeman field survives the inclusion of edge effects, which in the zigzag case induces a non-vanishing density of states at zero energy M.Fujita et al. (1996) for sufficiently wide ribbons Brey and Fertig (2006). We find results similar to those of Fig. 2 for doped graphene zigzag ribbons with edge Zeeman fields, with two differences: there are oscillations in the spin density of small magnitude as one moves in from the edge, and a small net spin is induced. We note that analogous spin configurations have been predicted to spontaneously form in ribbons when exchange interactions are important Son et al. (2006); our calculations demonstrate that such interactions are not needed to induce the tendency towards spin compensation. It is interesting to speculate that this effect might be utilized as a spin transfer device.
In summary, we have studied RKKY interactions among magnetic moments in graphene using a linear response approach. Our calculations show a strong qualitative difference between moments that couple symmetrically to the sublattices of the graphene honeycomb network and ones that couple to specific sublattices, with the latter showing more pronounced effects. Doped graphene in particular supports oscillations due to the Kohn anomaly only in the latter case. The sum of intra- and intersublattice responses was shown to vanish in the long wavelength limit in undoped graphene, leading to RKKY interactions of opposite sign for the two sublattices. Within mean-field theory, impurities coupled via these interactions should form a low temperature state analogous to that of a dilute antiferromagnet. Tight-binding calculations confirm the presence of the oscillations for doped graphene, and the tendency of opposite sublattices to have compensating spins.
After this work was completed, we became aware of related work Saremi () by S. Saremi on undoped graphene, which also concludes that the sign of RKKY interactions depends on whether moments are located on the same or opposite sublattices.
This work was supported by MAT2006-03741 (Spain) (LB), by the NSF through Grant No. DMR-0454699 (HAF), and by the US-ONR (SDS).
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