Introduction

In 1836, H. F. Talbot discovered that the intensity of light transmitted through a periodic grating exhibits a “self-imaging” of the grating at integer multiples of the distance away from the scattering array, where λ is the wavelength of light and d is the grating’s lattice constant1. This self-refocussing of the scattered light intensity is now referred to as the Talbot effect. As first explained by Lord Rayleigh2, the Talbot effect is the result of constructive interference of a coherent wave scattered from a periodic array. Within the realm of optical physics, the Talbot effect has been used in a variety of applications in nanolithography3, optical metrology and imaging4 and light field sensors5. The Talbot effect has also been observed in experiments on matter waves6, electron beams7,8, plasmonic devices9,10, wave guides and in photonic crystals11, along with a recent proposal12 to look at a spin Talbot effect in a two-dimensional electron gases (2DEG).

Sir Michael Berry was the first to make a deeper connection between the physics of the Talbot effect and that of quantum revivals observed for confined quantum particles13,14,15, where an initial quantum wave packet undergoes spatiotemporal refocussing as a result of quantum interference. With the discovery of new materials that possess electronic structures that can be described by the relativistic Dirac equation, such as monolayer graphene16 and the two-dimensional surface states of topological insulators17,18,19 such as Bi2Se3, theoretical extensions of the Talbot effect to the case of relativistic quantum revivals were also performed20,21,22 where it was shown that under certain conditions, bound relativistic particles with nonzero mass could also exhibit spatiotemporal revivals. From this theoretical work, however, it was not clear whether quantum revivals or, for that matter, the Talbot effect could exist for massless Dirac fermions (mDfs) since confining such particles is difficult due to Klein tunneling23,24. While recent numerical calculations25 have shown that a Talbot effect can be present in two-dimensional phononic crystals with a dispersion relation that mimics the mDf dispersion relation, a full theory of the Talbot effect for mDfs is still lacking.

In this paper, we consider the relativistic analogue of Talbot’s original experiment applied to a monochromatic beam of two-dimensional mDfs transmitted through a periodic one-dimensional potential. In order to place our theoretical results within a physically realizable context, we consider the particular case of intravalley multiple scattering in monolayer graphene26 from a periodic array of localized scatterers as illustrated in Fig. 1. Our previous theoretical work27 for the scattering of mDf waves from a one-dimensional periodic array of localized scatterers is generalized and used to demonstrate that a Talbot effect exists for mDfs. Furthermore, the effects of the mDfs’ spinor nature on the predicted Talbot effect is shown to generate an additional amplitude modulation and phase shift in the probability density that is most pronounced for mDfs at non-normal incidence to the scattering array.

Figure 1
figure 1

Scattering of an incident mDf wave in graphene with energy , , from a one-dimensional array of localized, cylindrically symmetric, nonmagnetic scatterers. In the Figure, the unit cell consists of Ns = 3 localized cylindrically symmetric scatterers. The positions of the scatterers are denoted by where the subscript denotes the particular scatterer in the nth unit cell.

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Results

We consider the case of a mDf wave in graphene with energy E = ħvFk1 ≥ 0 and wave vector , , that is incident to a one-dimensional array of localized, cylindrically symmetric, nonmagnetic scatterers as shown in Fig. 1. The subscript, , is the valley index and denotes the corresponding Dirac point that the scattering solutions are expanded about. The transmitted wave function to the right of the scattering array (x ≫ d) is given by:

The sum in equation (1) is over all “open” channels denoted by integers where and , where {z}+ corresponds to the smallest integer greater than z and {z}− corresponds to the largest integer less than z. For , the wave vector associated with the nth open channel, with and , is purely real. Note that for incident waves with wavelengths satisfying , and . Under these conditions, the incident wave is not scattered by the scattering array and is perfectly transmitted. In Supporting Information, general expressions for the transmission coefficients, Tn in equation (1), are provided.

Writing the transmission coefficient for the nth open channel as for n ≠ 0 and , the dimensionless probability density for x ≫ d, , can be written as:

where and . In equation (2), the interference between different open channels contained in in equation (1) will generate the Talbot effect for mDfs, which again requires that so that n ≠ 0 open channels are available to generate an interference pattern. Since the transmission coefficients are independent of the valley index or chirality of the incident wave, the probability density of the transmitted waves is also independent of the chirality of the incident waves.

For comparison, the dimensionless probability density for an achiral electron wave with an effective mass of m* in a 2DEG, , can be written as (see Supporting Information for details):

where for n ≠ 0 and are the transmission coefficients for the nth open channel with and . Comparing equations (2) and (3), the dimensionless probability densities for both mDfs and 2DEGs consist of a constant plus a sum over cosine terms that are periodic along both the - and -directions with periods and for m ≠ n, respectively. In particular, for a normally incident wave with λ < d, and the periodicity along the -direction in equation (2) and equation (3) can be used to define a set of “Talbot lengths”, for |m| > |n| ≥ 0 and , which are given by:

When λ ≪ d, the paraxial approximation gives . The traditional Talbot distance defined by Lord Rayleigh2 corresponds to . Thus the Talbot length for mDfs and achiral 2DEGs are identical to the traditional Talbot length. Furthermore, similar phase shifts, in equation (2) for a mDf and in equation (3) for an achiral 2DEG, are both the result of the scattering potential, which is reminiscent of the phase shifts associated with the Talbot-Beeby effect28. However, due to the spinor nature of the mDfs, the cosine terms in equation (2) possess an additional amplitude factor of and the last term in equation (2) also has an additional phase shift of that is due to the interference between the incident wave, and the open scattering channels.

Similarly, the reflected wave function, for x ≪ −d, is given by:

where for . Expressions for the reflection coefficients, in equation (5), are given in Supporting Information. The dimensionless probability density to the left of the scattering array, for x ≪ −d, is given by:

where . For comparison, a similar calculation of the dimensionless probability density in a 2DEG gives:

where are the reflection coefficients for the nth open channel with (expressions are given in Supporting Information) and . Comparing equations (6) and (7), the dimensionless probability densities for both mDfs and 2DEGs consist of a constant plus a sum over cosine terms that are periodic along both the - and -directions with periods either given by and for m ≠ n, respectively, or by and for n ≠ 0, respectively. Due to the spinor nature of the mDfs, however, there are again additional amplitude factors of that are identical to those found in equation (2) for the transmitted wave. Furthermore, the interference between the incident wave and the n ≠ 0 reflected waves results in an amplitude factor of along with a phase shift of relative to that found in an achiral 2DEG. As a result, a greater difference in the probability densities between mDfs and achiral 2DEGs will in general be observed for the reflected waves relative to that found for the transmitted waves.

In Fig. 2, numerical calculations of the dimensionless probability densities for a mDf and an achiral 2DEG wave normally incident to a one-dimensional array of scatterers with lattice constant d = 30 nm and with a unit cell consisting of a single scatter of potential V = −0.2 eV and radius rs = 4 nm are shown. For comparison, k1 was chosen to be the same in both the mDf and the achiral 2DEG in all cases; inside the scattering regions, the magnitude of the wave vector was chosen to be in both the mDf and the achiral 2DEG. In the plots of and in Fig. 2, the following values of k1d, lmax + 1 partial waves scattering from a single scatterer and the open scattering channels were used in the calculations along with the corresponding total transmission probabilities: (Fig. 2a) k1d = 3.1845π, lmax + 1 = 5, , and , (Fig. 2b) k1d = 8.1845π, lmax + 1 = 7, , Ttot = 0.9862 and and (Fig. 2c) k1d = 14.1845π, lmax + 1 = 10, , Ttot = 0.9936 and . As seen in Fig. 2, similar periodic patterns in the probability density appear to the right of the scattering array in both the mDf and achiral 2DEG calculations whereas the amplitude factors and phase shifts of found in the reflected mDf probability density in equation (2) lead to large differences in the probability to the left of the scattering array relative to that found in an achiral 2DEG [equation (6) vs. equation (7)]. The difference in probability densities between the mDF and an achiral 2DEG was most pronounced for the longest wavelength case shown in Fig. 2a due to (i) the larger difference in reflection probabilities between the mDf and achiral 2DEG cases ( in Fig. 2(a) versus ΔR = 0.0225 and ΔR = 0.0073 in Fig. 2(b,c), respectively) and (ii) the fact that the reflected probability is spread out over fewer backscattering channels in the longer wavelength case [n ∈ {±1}] relative to the shorter wavelength cases [n ∈ {±1, ±2, ±3, ±4} in Fig. 2b and n ∈ {±1, ±2, … ±6, ±7} in Fig. 2c].

Figure 2
figure 2

Plots of the dimensionless probability densities in a mDf, and in an achiral 2DEG, , for an electron wave normally incident to an infinite one-dimensional array with lattice constant d = 30 nm consisting of a single scatterer per unit cell with rs = 4 nm and V = −200 meV at the following wave vector magnitudes and lmax: (a) and lmax = 4, (b) and lmax = 6 and (c) and lmax = 9.

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It is known from previous theoretical26,29,30 and experimental31,32,33,34 work that a particle’s spinor nature can significantly affect the observed interference patterns of waves undergoing multiple scattering. However, the observed differences in the probability densities of an mDf and achiral 2DEG in Fig. 2 are due not only to the spinor nature of the mDfs but also due to differences in transmission and reflection coefficients, Tn and Rn for the mDf versus and for the achiral 2DEG. Therefore, to isolate the effects of the spinor nature of the mDfs on the probability density, we can replace and by Tn and Rn in the right hand sides of equation (3) and equation (7) to calculate the probability density for a “spinless” mDf, . In this case, the relative difference in probability density due solely to the spinor nature of the mDfs, χ, can be calculated using:

Plots of the dimensionless probability densities and relative probability density differences, χ [equation (8)], for both regular and “spinless” mDfs waves scattering from the same scattering potentials used in Fig. 2 are given in Fig. 3 (at normal incidence, ) and Fig. 4 (at non-normal incidence, ). Two different wave vector amplitudes were used in the calculations: k1d = 3.845π [Figs 3a and 4a] and k1d = 5.5π [Figs 3b and 4b]. At normal incidence (Fig. 3), the relative probability density difference to the right of the scattering array, which is mainly due to the cos(ϕspinor) amplitude factors in equation (2), is only significant over a small area. However, at non-normal incidence ( in Fig. 4), the probability densities are significantly different between the normal and “spinless” mDfs over a larger area, which is consistent with our theoretical predictions. In this case, the difference in probability density is due not only to the cos(ϕspinor) amplitude factors but also the phase shifts generated from the interference between the incident wave and the n ≠ 0 “open” transmitted/reflected waves in equations (2) and (6).

Figure 3
figure 3

Plots of the dimensionless probability densities in a regular mDf, and in a spinless mDf, (equations 3 and 7 with and ) along with their relative probability differences χ [equation (8)]. Calculations were performed for waves at normal incidence to an infinite one-dimensional array with lattice constant d = 30 nm consisting of a single scatterer per unit cell with rs = 4 nm and V = −200 meV at the following wave vector magnitudes, Ttot and lmax: (a) , Ttot = 0.9867 and lmax = 4 and (b) , Ttot = 0.9650 and lmax = 6.

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Figure 4
figure 4

Plots of the dimensionless probability densities in a regular mDf, and in a spinless mDf, (equations 3 and 7 with and ) along with their relative probability differences χ (equation (8)). Calculations were performed for waves at non-normal incidence to an infinite one-dimensional array with lattice constant d = 30 nm consisting of a single scatterer per unit cell with rs = 4 nm and V = −200 meV at the following wave vector magnitudes, Ttot and lmax: (a) , Ttot = 0.8627 and lmax = 4 and (b) , Ttot = 0.9485 and lmax = 6.

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While the results in Figs 2, 3, 4 considered a scattering array with a unit cell consisting of a single scatterer, the theory developed in this work can also be applied to arbitrary scatterer configurations within a unit cell. In Figs 5 and 6, was calculated for a wave with k1d = 5.5π that was normally incident () to a scattering array with lattice constant d = 30 nm and with a unit cell consisting of four scatterers at potential V = −0.33 eV that were either in a collinear arrangement with , , and as shown in Fig. 5 or in a nonlinear arrangement with , , and as shown in Fig. 6. For both scatterer arrangements, the sizes of the scatterers were taken either to be equal [rs1 = rs2 = rs3 = rs4 = 4 nm in Figs 5b and 6b] or unequal [rs1 = rs4 = 4 nm, rs2 = 2 nm and rs3 = 6 nm in Figs 5a and 6a]. The nonlinear arrangement of scatterers led to larger total transmission probabilities relative to the linear arrangement [Ttot = 0.5514 in Fig. 6a vs. Ttot = 0.3739 in Fig. 5a and Ttot = 0.4611 in Fig. 6b vs. Ttot = 0.232 in Fig. 5b]. For both types of scatterer arrangements, the total transmission probabilities were also larger when the scatterers were of unequal sizes. Finally, although the Talbot lengths, in equation (4), depend solely upon λ and d, the fine structure in depends sensitively upon the details of the scatterer sizes, potentials and arrangements within a unit cell, which ultimately determines the various transmission and reflection coefficients, Tn and Rn in equation (2) and equation (6), respectively.

Figure 5
figure 5

Plots of the dimensionless probability densities in a mDf, for a wave with k1d = 5.5π normally incident to an infinite one-dimensional array with lattice constant d = 30 nm and a unit cell consisting of Ns = 4 scatterers of potential V = −330 meV in a collinear arrangement with , , and . The scatter sizes were either chosen to be either (b) equal with rs1 = rs2 = rs3 = rs4 = 4 nm, which resulted in Ttot = 0.2320 or (a) unequal with rs1 = rs4 = 4 nm, rs2 = 2 nm and rs3 = 6 nm, which resulted in Ttot = 0.3739. In both calculations, lmax = 6 was chosen.

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Figure 6
figure 6

Plots of the dimensionless probability densities in a mDf, for a wave with k1d = 5.5π normally incident to an infinite one-dimensional array with lattice constant d = 30 nm with a unit cell consisting of Ns = 4 scatterers of potential V = −330 meV in a nonlinear arrangement with , , and . The scatter sizes were chosen either to be (b) equal with rs1 = rs2 = rs3 = rs4 = 4 nm, which resulted in Ttot = 0.4611, or (a) unequal with rs1 = rs4 = 4 nm, rs2 = 2 nm and rs3 = 6 nm, which resulted in Ttot = 0.5514. In both calculations, lmax = 6 was chosen.

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Finally, we consider an incident wave scattering from a finite scattering array. In this case, previous theoretical work on multiple scattering from a finite number of scatterers26,27 was applied to calculate . In Fig. 7, for a wave normally incident to a finite scattering array is shown, where the scattering array consists of N = 21 equally spaced cylindrically symmetric scatterers with rs = 4 nm and V = −200 meV that were placed along the -axis between to with d = 30 nm. The incident wave vectors were chosen to be identical to those used in Fig. 2 to enable a better comparison of between the finite and infinite scattering arrays. While the overall periodic structures observed in were similar in both the finite [Fig. 7] and infinite [Fig. 2] cases, some of the finer structures/interference patterns observed in the infinite scattering array were absent for the finite scattering array. The periodic structures in the finite case also became blurrier with increasing distance from the scattering array, particularly at distances x ≫ 10d from the center of the scattering array. This was a consequence of the finite size of the scattering array whereby the interference patterns in decay approximately as . However, at distances within from the center of the scattering array, a clear periodic pattern was still observed in the case of a finite scattering array.

Figure 7
figure 7

Plots of for a normally incident mDf wave to a finite one-dimensional array of 21 identical scatterers with rs = 4 nm, V = −200 meV and with the position of the nth scatterer given by with d = 30 nm and n ∈ [−10, 10]. The same wave vectors and lmax values used in Fig. 2 were also used for the finite scattering array: (a) and lmax = 4, (b) and lmax = 6 and (c) and lmax = 9.

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Discussion

In this work, the theory of the two-dimensional Talbot effect for massless Dirac fermions (mDfs) was presented. It was shown that the Talbot effect for mDfs exists with Talbot lengths, in equation (4), that were identical to those found for an achiral two-dimensional electron gas (2DEG). The interference patterns seen in the Talbot effect are a result of coherent electron transmission of mDfs through the scattering array, whereby multiple scattering pathways constructively interfere at distances away from the scattering array determined by the periodicity of the scattering array. However, due to the spinor (or pseudospinor in the case of graphene) nature of mDfs, the periodic structures found in the probability density were both amplitude modulated and phase shifted relative to those found in an achiral 2DEG. Such differences were most pronounced for mDf waves at non-normal incidence to the scattering array. Numerical calculations on finite scattering arrays demonstrated that periodic structures in the probability density still exist but that these structures decay with increasing distance from the scattering array. While the probability density is independent of which valley point the scattering states are expanded about [equations (2) and (6)], the use of magnetic scatterers could potentially be used to distinguish the chirality (or in this case, valley index ) of the incident waves in monolayer graphene. The mDf Talbot effect predicted in this work should be observable in systems like monolayer graphene and on the surfaces of topological insulators, where phase coherence lengths greater than 5 μm and 1 μm have been experimentally observed in graphene35 and topological insulators36, respectively. Overall, this work provides yet another example of the fruitful analogy between traditional optics and coherent “electron” optics in graphene and similar systems37,38,39,40.

While there exist proposals9,10 to employ the Talbot effect for nonrelativistic electrons in plasmonic devices, the theory presented in this work could be used as a starting point for designing and understanding the Talbot effect in graphene and topological insulator41,42 plasmonic devices. It should also be noted that only coherent dynamics was considered in this work. Spatial and spin/pseudospin decoherence, however, will attenuate and destroy the Talbot effect with increasing distance from the scattering array. As a result, comparing the observed spatial decay of the interference patterns in the Talbot carpet with the interference patterns calculated using the theory presented in this work could provide valuable information about both spatial decoherence43 and spin/pseudospin decoherence in two-dimensional mDfs.

Methods

The basic results for intravalley scattering of a plane wave incident to a one-dimensional array of localized scatterers in graphene (as illustrated in Fig. 1) and in a 2DEG are derived in Supporting Information26,27,44,45,46. The overall theoretical formalism used in this paper represents a generalization of the case of a single scatterer per unit cell27 to the case of multiple scatterers within a unit cell. From Fig. 1, the incident waves with energy , which are labeled by the corresponding valley index or Dirac point that the plane wave states are expanded about in graphene, , are given by , where with for waves incident to the scattering array from the left. The unit cell of the scattering array consists of Ns localized cylindrically symmetric scatterers with a lattice constant d. The overall scattering potential can be written as where denotes the position of the mth scatterer in the nth unit cell, Vm and rsm are the potential and radius of the mth scatterer, respectively and is Heaviside step function given by:

In this work, the potentials of the individual scatterers were taken to be identical in order to avoid the confounding effects of electric fields between the scatterers, i.e., Vm = V for all m ∈ {1, 2, …, Ns}. The lth partial wave scattering amplitude from the mth cylindrically symmetric scatterer, sm,l with , is given by26,46:

where is the magnitude of the wave vector inside scatterer regions and Jl(z) is a bessel function of the first kind of order l, respectively. For the nth scatterer with lmax,n + 1 partial waves were chosen to account for greater than 99.9999% of the total scattering amplitude, i.e., . For Ns scatterers within a unit cell, lmax is just the maximum partial wave needed to take into account at least 99.9999% of the total scattering amplitude from all scatterers, i.e., . Derivations of the scattering solutions for both a mDf and a 2DEG are given in Supporting Information. Finally, all calculations shown in Figs 2, 3, 4, 5, 6, 7 were carried out using in-house MATLAB (Mathworks) programs.

Additional Information

How to cite this article: Walls, J. D. and Hadad, D. The Talbot Effect for two-dimensional massless Dirac fermions. Sci. Rep. 6, 26698; doi: 10.1038/srep26698 (2016).