Straintronics beyond homogeneous deformation

We present a continuum theory of graphene, treating on an equal footing both the homogeneous Cauchy-Born (CB) deformation and the microscopic degrees of freedom associated with the two sublattices. While our theory recovers all extant results from homogeneous continuum theory, the Dirac-Weyl equation is found to be augmented by new pseudogauge and chiral fields fundamentally different from those that result from homogeneous deformation. We elucidate three striking electronic consequences: (i) non-CB deformations allow for the transport of valley-polarized charge over arbitrarily long distances, e.g., along a designed ridge; (ii) the triaxial deformations required to generate an approximately uniform magnetic field are unnecessary with non-CB deformation; and finally (iii) the vanishing of the effects of a one-dimensional corrugation seen in ab initio calculation upon lattice relaxation is explained as a compensation of CB and non-CB deformation.

Figure 1

Functional relationship between deformation and effective fields for Cauchy-Born and non-Cauchy-Born deformations. Shown are (1a) in-plane acoustic, (2a) in-plane optical, (3a) out-of-plane acoustic, (4a) out-of-plane optical deformation fields with the corresponding effective fields the gauge field (second column), the pseudomagnetic field (third column), and the scalar field (fourth column). The scalar field is an electric potential for the Cauchy-Born deformation and a chiral (mass-generating) potential for the non-Cauchy-Born deformation. Note that the effective fields in row 4 result from the combination of the deformation fields in (3a) and (4a); i.e., these effective fields result from a coupling of the out-of-plane acoustic and optical deformation fields. For the Cauchy-Born deformations (rows 1 and 3) the pseudomagnetic field displays the clear influence of the underlying graphene lattice in the 6-fold petal structure of the field. In dramatic contrast the pseudomagnetic field for the non-Cauchy-Born deformations follows closely the structure of the deformation field itself, and exhibits no symmetry lowering due to the underlying lattice. For further details on the nature of the deformations and the resulting effective fields, see Sec. 4b. Calculations are performed at the K valley of the graphene BZ, and the sample area is $700a\times 700a$ with $a$ the lattice constant of graphene. At the $K^{*}$ valley the sign of the pseudogauge is reversed.

Figure 2

Percentage change in the trace of the Fermi velocity tensor due to deformation for the four deformation types shown in Fig. 1. (a) In-plane acoustic deformation; (b) In-plane optical deformation; (c) Out-of-plane acoustic deformation; (d) Coupling of out-of-plane optical and acoustic deformations. Both of the optical deformations, panels (b) and (d), generate distinctly different changes in the local Fermi velocity as compared to the acoustic type deformations, panels (a) and (c).

Figure 3

Sublattice projected local density of states for the four types of deformations shown in Fig. 1. The acoustic (i.e., homogeneous Cauchy-Born) deformations generate “charge flowers” with three of the “petals” exhibiting strong charge localization on the A sublattice with the other three petals strongly localized on the B sublattice. In dramatic contrast, the optical (i.e., non-Cauchy-Born) deformation generates “charge walls” and “charge dots” in which the A sublattice is localized at the perimeter of the deformation, and the B sublattice in the interior of the optical deformation field. In either case, the charge densities follow closely the pattern of the pseudomagnetic fields induced by the deformation; see Fig. 1. Results are obtained by integrating an energy window from 0 to 100 meV; similar features are found in any energy window close to the Dirac point. Calculations are performed at the K valley of the graphene BZ.

Figure 4

Current densities due to homogeneous Cauchy-Born obeying deformations (acoustic) and non-Cauchy-Born deformations (optical), displaying the striking difference between current structure from these two types of deformations. Shown are current densities for all six possible cases of deformations: (a) acoustic in-plane, (b) optical in-plane, (c) optoacoustic in-plane, (d) acoustic out-of-plane, (e) optical out-of-plane, and (f) optoacoustic out-of-plane. Acoustic deformations give rise to local current loops [46, 48], 3 clockwise and 3 anticlockwise for a centrosymmetric deformation, but with more complex structure for the noncentrosymmetric cases, as seen in panels (a) and (d). In contrast, optical deformations, or deformations with an optical component, generate large-scale current loops following the structure of the deformation, panels (b), (c), and (f). In all cases the underlying physical mechanism is snake states along nodal lines of the pseudomagnetic field. The exception is the out-of-plane optical deformation, which is purely imaginary (see term 11 in Table 2) and generates almost no current (reduced by two orders of magnitude as compared to all other cases), and exhibits almost no discernible connection to the deformation field. We attribute these small currents to the Fermi velocity anisotropy that results in a nonzero imaginary pseudomagnetic field (see Sec. 4b). Calculations are performed at the K valley of the graphene BZ, at the $K^{*}$ valley the sign of all currents is interchanged, as required by time-reversal symmetry. The energy window of integration is between 0 and 100 meV, with similar results found in any energy window close to the Dirac point.

Figure 5

Impact of inclusion of $\sigma$ hopping on the coupling between in- and out-of-plane deformations. Panels (a) and (b) display the in-plane and out-of-plane deformation fields, and panel (c) the resulting pseudomagnetic field. In panels (d)–(f) are shown, respectively, the sublattice projected density and the current, integrated in an energy window from 0 to 100 meV at the K valley (similar results are found at other energy windows close to the Dirac point). As can be seen by comparison with Figs. 1, 3, and 4, which include only $\pi$ hopping, the qualitative physics of the pseudogauge following the deformation is unchanged. The sample size is $700a\times 700a$ with $a$ the lattice constant of graphene.

Figure 6

The deformation field employed in the nonlinear Gaussian bump of Fig. 7, which consists of (a) an out-of-plane acoustic deformation as a Gaussian bump of height around 40 Å and (b) an in-plane optical deformation in the form of a Gaussian ring (obtained as the scaled derivative of the acoustic field). The sample size is $700a\times 700a$ with $a$ the lattice constant of graphene.

Figure 7

The nonlinear Gaussian bump. Shown are the (a) pseudomagnetic fields, (b) current densities, and projected charge densities on the A sublattice (c) and the B sublattice (d) for a Gaussian bump with only the acoustic Cauchy-Born deformation (first row), but in the second and third rows with the addition on an in-plane optical deformation of maximum magnitude ${10}^{-3}\AA$ (second row) and $7\times {10}^{-3}\AA$ (third row). The Cauchy-Born deformation field is shown in Fig. 6, with the optical non-Cauchy-Born component for the last row in Fig. 6. Evidently, the inclusion of an optical component in the deformation field has a dramatic effect on all physical quantities, with notably the 6-fold local current pattern of the Cauchy-Born deformation replaced by a single current density circulating around the deformation, and a striking change in the pseudospin polarization from the “charge flowers” seen in panels (1c) and (1d) to a polarization between the perimeter and interior of the deformed region seen in panels (3c) and (3d). The energy window of integration is 0 to 100 meV, but similar results are found in any energy window close to the Dirac point. The sample size is $700a\times 700a$ with $a$ the lattice constant of graphene.

Figure 8

Optical quenching of armchair corrugations in graphene: In panel (a) the acoustic and optical deformation fields are shown, see Eq. (51) and Eq. (53), respectively, representing a corrugation of the graphene lattice in the armchair direction. The acoustic deformation is taken from Ref. [29]. (b) The acoustic field generates a gauge field that displaces the Dirac cone off the high-symmetry K point, dark (black) symbols, while addition of the optical deformation field exactly compensates this gauge resulting in the return of the Dirac cone to the K point, light (red) symbols, thus quenching the effect of the acoustic deformation. (c) The density of states (DOS) for a corrugation period of 170.4 Å, showing that optical quenching in addition can remove Landau levels that would be seen in the absence of the optical deformation field. (d) Optical deformation can by itself create Landau levels; shown is a $700a$ corrugation with $a$ the lattice constant of graphene (in the zigzag direction) resulting in a clear sequence of chiral Landau levels ($E_n\propto \sqrt{n}$, $n\in \mathbb{N}$).