Analog of Fishtail Anomaly in Plastically Deformed Graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l JETP Letters
Analog of Fishtail Anomaly in Plastically Deformed Graphene
S. Sergeenkov and F.M. Araujo-Moreira
Grupo de Materiais e Dispositivos, Departamento de F´ısica, Universidade Federal de S ao Carlos, 13565-905 S ao Carlos, SP, Brazil
By introducing a strain rate ˙ ǫ generated pseudo-electric field E dx ∝ ~ ˙ ǫ , we discuss a magnetic responseof a plastically deformed graphene. Our results demonstrate the appearance of dislocation induced param-agnetic moment in a zero applied magnetic field. More interestingly, it is shown that in the presence of themagnetoplastic effect, the resulting magnetization exhibits typical features of the so-called fishtail anomaly.The estimates of the model parameters suggest quite an optimistic possibility to experimentally realize thepredicted phenomena in plastically deformed graphene.PACS: 81.05.ue, 62.20.F-, 75.80.+q
1. Introduction.
Many interesting and unusualphenomena due to significant modifications of the car-bon based materials (including graphene) under me-chanical deformations leading to generation of strongintrinsic pseudomagnetic fields have been recently dis-cussed (see, e.g. [1, 2, 3, 4, 5] and further referencestherein). Of special interest for us are dislocations re-lated properties in plastically deformed graphene andcarbon nanotubes [6, 7, 8, 9].In this Letter we consider theoretically some intrigu-ing magnetic properties of graphene sheet under plasticdeformations by directly incorporating a constant strainrate as a time dependent gauge potential into the Diracmodel. The physics behind our findings and feasibilityof their experimental verification are discussed.
2. Model.
Recall [1, 2, 5] that in the absence ofchirality (intervalley) mixing, the low-energy electronicproperties of graphene near the Fermi surface can bereasonably described by a two-component wave func-tion | Ψ > = (Ψ , Ψ ) obeying a massless Dirac equation i ~ ∂ | Ψ >∂t = H| Ψ > (1)with an effective Hamiltonian H = v F ( σ x π x + σ y π y ) (2)Here, π a = p a + eA a + eA da with p a = − i ~ ∇ a beingthe momentum operator, A a the electromagnetic vectorpotential, and A da the deformation induced vector po-tential; σ a are the Pauli matrices, and v F is the Fermivelocity. In what follows, a = { x, y } . Let us considera graphene sheet of length L and width W under thesimultaneous influence of plastic deformation and per-pendicular applied magnetic field B z (defined via thevector potential A y = B z x ). As is well known [1, 2, 5],the homogeneous strain ǫ induced gauge potential, givenby eA dx = ~ ǫ/r (where r = 0 . nm is carbon-carbon bond length), leads to appearance of intrinsic pseudo-magnetic field B dz = ~ ǫ/eLr inside deformed graphenelattice. By analogy with an applied electric field E x (de-fined via time-dependent vector potential A x = E x t ),we introduce plastic deformation effects into the modelthrough a constant plastic strain rate ˙ ǫ dependent vectorpotential A dx = E dx t resulting in appearance of intrinsicpseudo-electric field E dx = ~ ˙ ǫ/er .
3. Results and Discussion.
Let us considerthe magnetic response of the graphene sheet (with area S = LW ) on plastic deformation by analyzing its mag-netization: M z ( B z , ˙ ǫ ) ≡ − S (cid:20) ∂ E ( B z , ˙ ǫ ) ∂B z (cid:21) (3)Here E ( B z , ˙ ǫ ) = Z τ dtτ Z L dxL Z W dyW X i =1 h Ψ i |H| Ψ i i (4)is the total energy of the problem based on the pre-viously obtained [5] solutions | Ψ i > of time-dependentEq.(1) ( τ is the characteristic time related to durationof plastic deformation, that is ˙ ǫ ≃ ǫ/τ ). First of all,the analysis of Eqs.(1)-(4) reveals that plastic deforma-tion results in appearance of a non-zero magnetic mo-ment µ z = M z (0 , ˙ ǫ ) S = ~ v F τ ˙ ǫ/r in a zero applied mag-netic field ( B z = 0). For typical experimental values ofthe applied strain rates [6, 9] ˙ ǫ ≃ − s − , we obtain µ z ≃ µ B for a reasonable estimate of the plastically in-duced paramagnetic moment in graphene [10] ( µ B is theBohr magneton). Fig. 1 shows the field dependence ofthe induced magnetization ∆ M z = M z ( B z , ˙ ǫ ) − M z (0 , ˙ ǫ )for different values of the normalized strain rate ˙ ǫ where B = Φ /S is a characteristic magnetic field (Φ is theflux quantum). It is worth noting that it closely followsthe observed [10] behavior of the point defects inducedparamagnetic moment in graphene for different values1 D M z / M B z /B Fig.1. The magnetic field dependence of the normalizedmagnetization in plastically deformed graphene for dif-ferent values of the normalized strain rate (from bot-tom to top): ˙ ǫ/ ˙ ǫ = 0 . , . , . , and 1 .
0. Here,˙ ǫ = 10 − s − . of density of vacancies ρ v . This makes sense becauseplastic deformation is driven by motion of dislocations(with velocity v d ) leading to strain rate dependence onboth v d and dislocation density ρ d as follows, ˙ ǫ = bρ d v d .Here b is the absolute value of the relevant Burgers vec-tor. It is instructive to point out that the dislocationvelocity v d in a sense plays a role of the Fermi veloc-ity v F (which links applied electric and magnetic fieldsas E = v F B ) in relationship between strain rate in-duced pseudo-electric E d and strain induced pseudo-magnetic B d fields. Indeed, with quite a good accu-racy we can write E d ≃ v d B d . At the same time, itis important to emphasize that (in addition to a defi-nitely non-universal character of v d ) these two charac-teristic velocities describe phenomena on a completelydifferent scale because while v F ≃ m/s , typical dis-location velocities rarely exceed v d ≃ − m/s . Letus turn now to another interesting phenomenon relatedto behavior of plastic deformation under applied mag-netic field. Namely, as it was experimentally observedfor many different types of materials (including semi-conductors), upon application of magnetic field (of theorder of B = 1 T ), dislocation velocity v d (and hencestrain rate) becomes strongly field dependent. This phenomenon is called a magnetoplastic effect (MPE).It was discovered by Al’shits et al in 1987 [11]. Thereare many different mechanisms which could be responsi-ble for such a behavior [12]. One of them (and probablymost appropriate for graphene [10]) is based on interac-tion between uncompensated spin of dislocation’s coreand point paramagnetic impurities [13, 14] due to thedifference in gyromagnetic factors g (so-called ∆ g mech-anism) leading to appearance of resonance frequency ω r = ∆ gµ B B/ ~ in applied magnetic field B . As a result,the interaction energy U between dislocation and impu-rity becomes field dependent with U ( B ) > U (0), whichin turn leads to a significant increase of the thermallyactivated dislocation velocity described by the followingexpression [13, 14] v d ( B ) = v d (0) exp (cid:20) ∆ U ( B ) k B T (cid:21) (5)where ∆ U ( B ) = U ( B ) − U (0) = U (0) f ( B ) (6)with f ( B ) = B B + B p (7)Here, B p = ~ / ∆ gµ B τ s is the characteristic field formanifestation of MPE with τ s ≃ ω − r ( B = B p ) beingthe characteristic time. For typical values of ∆ g ≃ − and τ s ≃ − s , we get B p ≃ T for the estimateof the intrinsic magnetic field due to spin-mediated in-teraction between point and linear defects [11, 12, 13].So far, we have ignored the MPE in the magnetic re-sponse of graphene under plastical deformation. Letus see now what happens with magnetization in thepresence of the above discussed MPE, that is assumethat the strain rate becomes field dependent as follows,˙ ǫ ( B z ) = bρ d v d ( B z ) with v d ( B ) given by Eq.(5). No-tice that accounting for MPE virtually transforms ourpseudo-electric field E dx into a pseudo-magnetoelectricone E dx ( B z ). The obtained magnetic field dependenceof the resulting magnetization is shown in Fig. 2 for U (0) = 0 . k B T , ˙ ǫ (0) = 10 − s − and for different val-ues of the ratio γ = B p /B between two characteristicfields (notice that in Fig. 2 the applied field is normal-ized to B p instead of B as in Fig. 1). We observea remarkable fishtail like behavior of magnetization inplastically deformed graphene in the presence of MPE(a ”diamagnetic” part of the curve − M z /M in Fig. 2is added for better visual effects only). As it is clearlyseen, the curve first reaches minimum at B z /B p ≃ B z /B p ≃ M z / M B z /B p g =0.5 g =0.3 g =0.1U(0)=0.01k B T Fig.2. The magnetic field dependence of the normalizedmagnetization in the presence of magnetoplastic effectfor U (0) = 0 . k B T , ˙ ǫ (0) = 10 − s − and three val-ues of γ = B p /B . A ”diamagnetic” part of the curve − M z /M is added for better visual effects only. attributed to a perfect match between the sizes of thevortex core and the pinning center. While for Abrikosovvortices the best pins are point defects (vacancies) [15],the so-called Josephson vortices (fluxons) require linear(or even planar) defects for their effective pinning [16].By analogy, we can assume that the discussed here MPEinduced fishtail anomaly in graphene structure probablyhas something to do with a perfect match (energy min-imization) between a paramagnetic impurity and mag-netic field modified dislocation, which serves as a spin-sensitive pinning center for this impurity [7, 8, 10]. Fortypical values of the width W = 40 nm and aspect ra-tio L/W = 10 we obtain B = Φ /W L ≃ T for theestimate of the characteristic field in graphene (shownin Fig. 1) which should be compared with the earlierestimated value of the MPE mediated intrinsic mag-netic field B p ≃ T . According to Fig. 2, the fishtaillike behavior is expected to manifest itself already for γ = B p /B ≥ .
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