1. Introduction

In relativistic quantum mechanics, a potential barrier can become nearly transparent to an incoming particle if the potential exceeds the particle’s mass, in stark contrast to the non-relativistic quantum mechanics where a particle cannot transmit such a high potential barrier. This counterintuitive result has been known as Klein’s paradox [1,2].

To be explicit, we consider the case that a relativistic particle with mass $m$ and energy $E>0$ is transmitted from a region without potential (Region I) into a potential barrier $V\geq 0$ (Region II), as shown in Fig. 1(a). The transmission is categorized into three regimes, which we call the small-$V$ regime ($E\geq V$), the reflected regime ($E<V<E+mc^2$), and the large-$V$ regime ($V\geq E+mc^2$). In the small-$V$ regime, the particle is transmitted similarly to the non-relativistic quantum tunneling (Fig. 1(b)). In the reflected regime the particle is fully reflected. Most interestingly, in the large-$V$ regime, the particle is transmitted as an antiparticle (Fig. 1(c)).

 

Fig. 1. (a) Schematic of the transmission process: A relativistic particle in Region I with mass $m$ and energy $E$ is transmitted into a potential barrier $V$ (Region II). $\psi ^\mathrm {in}$, $\psi ^\mathrm {r}$, and $\psi ^\mathrm {t}$ denote the incident, reflected, and transmitted wavefunctions, respectively. The way of transmissions depends on the value of the potential, namely $V<2mc^2$ (small) or $V>2mc^2$ (large), and the type of PhC in Region II (trivial or topological). (b) Conventional (normal) tunneling at small $V$. The red curve is the “particle” band and the blue curve is the “antiparticle” band. The black, blue, and orange dots correspond to the incident, reflected, and transmitted states in (a), respectively. The horizontal light-green line indicates that energy is conserved in the transmission process. (c) Massive Klein tunneling at large $V$. (d), (e) Band inversion occurs in topological PhC with negative mass, which allows Klein tunneling at small $V$ at the trivial-topological interface.

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Although Klein’s paradox has a simple resolution as shown in Fig. 1(c), the physical interpretation of Klein tunneling seems still under debate [2–11]. The large potential energy ($V>2mc^2\approx 1$MeV for electron) has led to theoretical interpretations of the paradox in terms of electron-positron pair creation using quantum field theory or quantum electrodynamics, which meanwhile makes its direct verification challenging in experiments of elementary particle physics. So far, Klein tunneling has been reported experimentally with massless particles in various condensed-matter systems [12–20], where there is no strict distinction between particles and antiparticles. In addition, these experiments consider the dispersion near the $\mathrm K$ and $\mathrm K'$ points with finite momenta instead of the $\Gamma$ point, which cannot be considered as ideal platform to clarify the Klein physics in a complete way. Klein tunneling was also studied in (but not limited to) deformed hexagonal lattices [21,22], photonic crystals [19,20,23,24], and magnonic systems [25], but to the best of our knowledge a direct observation of massive Klein tunneling is still missing.

In this article, we propose that honeycomb-type photonic crystals (PhCs) are ideal systems for investigating massive Klein tunneling. These systems possess doubly degenerate relativistic dispersions near the $\Gamma$ point [26–31]. So, electromagnetic modes in these systems behave as massive Dirac quasiparticles with four-component spinor wavefunctions. Recipes of PhC design giving quasiparticles with positive mass (trivial PhC), massless (photonic graphene), and even negative mass (topological PhC) have been established [26–31]. The advantage of these PhC systems is that the photonic band gap (mass gap) is on the order of $0.1$eV, which can be realized by semiconductor nanofabrication. We propose that an analog of massive Klein tunneling without potential can appear at the interface of PhCs with positive and negative mass (Fig. 1(d), (e)). Recent studies have shown that the transmission of light from air into one-dimensional PhCs depends on their topological properties [32–34]. Here, we show that interfaces between two-dimensional PhCs with opposite masses allow us to investigate the essential difference between normal and Klein tunneling.

This article is organized as follows. In Section 2, we explain how photonic eigenstates can be described as massive Dirac quasiparticles. In Section 3, we present our model of PhC interface and confirm massive Klein tunneling at the trivial-trivial PhC interface. We reveal that when the particle has a normal incidence, the transmission coefficient through a trivial-trivial PhC interface with a large/small $V$ is identical to that of a trivial-topological interface with a small/large $V$. In Section 4, we consider the way of refraction at PhC interfaces and find that transmission with a negative index of refraction is achieved in the large-$V$ regime, both for the trivial-trivial and trivial-topological interfaces. In Section 5, we investigate whether topological interfacial states [30,35,36] disturb the transmission process. In Section 6, we discuss the implications of our results.

2. Trivial and topological photonic crystals

Let us consider a honeycomb-type PhC made of dielectric pillars with unit cells containing six pillars as shown in Fig. 2(a), which was proposed in Ref. [26,27]. This system can be described by the tight-binding Hamiltonian

The photonic eigenstates near the $\Gamma$ point can be obtained from the $\mathbf {k}\cdot \mathbf {p}$ expansion with the basis $[p_+,d_+,p_-,d_-]$ [26,27]

These matrices satisfy the following anti-commutation relations

The photonic dispersion is shown in Fig. 2(c), where the mass gap is equal to $2M$. The blue curves are obtained by solving Eq. (2) numerically and the red curves show the Dirac dispersion $E=\pm \sqrt {A^2(k_x^2+k_y^2)+M^2}$ which is obtained by solving Eq. (8) with the plane-wave solution $\psi _+(\mathbf {r})=\psi _+ e^{i\mathbf {k}\cdot \mathbf {r}}$. The blue and red curves coincide near the $\Gamma$ point, which implies that photonic eigenstates can be described as massive Dirac quasiparticles.

 

Fig. 2. (a) PhC with a honeycomb lattice and hexagonal unit cells which contain six sites. The trivial and topological PhCs are realized by changing the hopping integrals inside and between the unit cells. (b) Photonic eigenstates at the $\Gamma$ point. (c) Photonic dispersion for a trivial PhC with $t_0=1.1t_1$, $M=t_0-t_1$ and $\Delta =2M$. The blue curve is obtained from the tight-binding Hamiltonian Eq. (1), while the red curve is obtained from the Dirac Hamiltonian Eq. (3).

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From its definition, it is clear that $M$ can be either positive or negative depending on the values of $t_0$ and $t_1$. A trivial PhC is obtained when $t_0>t_1$ (i.e. $M>0$). On the other hand, a topological PhC is obtained when $t_0<t_1$ (i.e. $M<0$). The negative mass for the topological PhC leads to band inversion, namely exchanging the $|{d_+}\rangle$ and $|{p_+}\rangle$ eigenstates in the order of energy near the $\Gamma$ point [26,27]. In other words, the positive and negative energy states, which correspond to “particles” and “antiparticles”, are inverted in a topological PhC.

In the above, we have shown how photonic eigenstates in honeycomb-type PhC can be described as massive Dirac quasiparticles with positive and negative masses. In what follows, we use such photonic eigenstates to study massive Klein tunneling at PhC interfaces. Being able to tune the Dirac mass and potential separately in the common platform is ideal for investigating the interplay between the Dirac mass and potential $V$ in experiments of Klein tunneling.

3. Transmission at PhC interface

3.1 Model

We consider the transmission of light through the interface of two PhC regions, namely Region I in $x<0$ and Region II in $x>0$ with a potential difference (Fig. 1), which can be achieved by changing the effective permittivity of the PhC in Region II. The interface in Fig. 2(a) along the $y$-direction is of zigzag type. Since we consider eigenstates near the $\Gamma$ point with wavevectors close to zero, where the photonic crystal can be approximated as a uniform structure, the interface shape does not affect the massive Klein tunneling under concern. The two PhCs have different $M$ and $A$ values, and the system close to the PhC interface is described by the following Hamiltonian

We use this Hamiltonian to solve the following eigenvalue equation

Using $\mathbb {1}\psi _+(\mathbf {r})=\psi _+(\mathbf {r})$ and Eq. (11), we arrive at the following decoupled differential equation

We use Eq. (12) to calculate the eigenvalues while Eq. (11) to calculate the eigenstates. For the transmission problem in Fig. 1, the following plane-wave solution is considered

Here, $\mathbf {k}^\mathrm {in}=(k_x^\mathrm {in},k_y^\mathrm {in}), \mathbf {k}^\mathrm {r}=(k_x^\mathrm {r},k_y^\mathrm {r})$ and $\mathbf {k}^\mathrm {t}=(k_x^\mathrm {t},k_y^\mathrm {t})$ are the wavevectors of the incident, reflected and transmitted wavefunctions, respectively. By definition of reflection we have $\mathbf {k}^\mathrm {in}\ne \mathbf {k}^\mathrm {r}$ (see Section 4 for more detail.) We solve Eq. (12) in each of the two regions separately and obtain

The continuity of the wavefunction at the interface ($x=0$) implies

Moreover, from Eq. (11) with Eq. (14), we obtain

3.2 Transmission coefficient

To discuss the transmission and reflection properties quantitatively, we calculate the conserved current $\mathbf {j}^\mu =(j_x^\mu,j_y^\mu )$ which is obtained from the continuity equation

The reflection coefficient $R$ and the transmission coefficient $T$ are calculated from the conserved currents:

It is clear that the condition $R+T=1$ is fulfilled. Although expressions similar to Eq. (26) and Eq. (27) have been obtained in literature [9,14], our results apply for both positive and negative masses.

Figure 3(a) and (b) show the transmission coefficient $T$ for a trivial-trivial interface and a trivial-topological interface, respectively, with $|M^<|=|M^>|=0.1t_0$ and $\phi ^\mathrm {in}=0$. The horizontal gray stripe shows the bandgap $\Delta =2\lvert M^<\rvert$ of the PhC in Region I, i.e. the region without incident particles. The blue stripe is the region with total reflection where $E$ lies within the bandgap of the PhC in Region II. Figure 3(c) and (d) show the line profile of $T$ at different $E$ values, namely $E=0.55\Delta$ and $E=\Delta$. The blue curve is the tunneling through the trivial-trivial interface, where the large-$V$ regime ($V>E+\Delta /2$) is identical to the Klein tunneling known so far [1,2]. The orange curve is the tunneling through the trivial-topological interface, which is strikingly different from the blue curve.

 

Fig. 3. Transmission coefficient $T$ at normal incidence $\phi ^\mathrm {in}=0$ for the trivial-trivial interface (a) and for the trivial-topological interface (b). Here, the bandgap in Region I ($-0.5\Delta \leq E\leq 0.5\Delta$ shown by the horizontal gray stripe) and the bandgap in Region II (blue stripe with $T=0$) are both given by $\Delta =2M$ with $M=|M^<|=|M^>|=0.1t_0$. (c) and (d) Line profile of $T$ at $E=0.55\Delta$ and $E=\Delta$. Klein tunneling and normal tunneling are assigned as in Fig. 1. Line profile of $T$ at constant $V$ values (vertical dashed lines in (a) and (b)) is shown in Fig. 4.

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To understand the difference between tunnelings at the trivial-trivial and trivial-topological interfaces, we observe that the kinematic factor ${\eta }$ can be written in the following form

We emphasize once again that, if $M^<$ and $M^>$ are both positive, Eq. (28) agrees with that in Ref. [2]. However, our result applies for both positive and negative masses. The trivial-trivial and the trivial-topological interfaces can be compared by introducing the effective mass $M_{\mathrm {eff}}=\mathrm {sgn}(E-V)M^>$. For the trivial-trivial interface, we have $M_{\mathrm {eff}}>0$ in the small-$V$ regime and $M_{\mathrm {eff}}<0$ in the large-$V$ regime. In contrast, for the trivial-topological interface, we have $M_{\mathrm {eff}}<0$ in the small-$V$ regime and $M_{\mathrm {eff}}>0$ in the large-$V$ regime. In other words, the effective mass is positive for the normal tunneling and negative for the Klein tunneling. Therefore, we conclude that the mass sign of the transmitted particle interchanges normal tunneling and Klein tunneling at the trivial-trivial interface and at the trivial-topological interface.

In the rest of this section, we investigate the tunneling at fixed $V$ values, which is close to real experimental setups. Figure 4 shows the tunneling through the trivial-trivial interface (blue curve) and through the trivial-topological interface (orange curve). Normal tunneling and Klein tunneling are identified as in Fig. 3. At $V=0$ (Fig. 4(a)), the current is fully transmitted at the trivial-trivial interface for any $E\geq \Delta /2$: This result is expected because there is no interface at $V=0$ and the energy spectrum is identical in Region I and Region II. On the other hand, for the trivial-topological interface at $V=0$, the current is fully reflected at the band edge $E=\Delta /2$: This is due to different parities at the $\Gamma$ point induced by band inversion. (This reflection mechanism has been used to construct topological cavity surface emitting lasers [37].) Then, the current is partially transmitted for $E>\Delta /2$ due to the hybridization of $|{d}\rangle$ and $|{p}\rangle$ eigenstates. The transmission changes when $\Delta >V>0$ (Fig. 4(b)) because the states in Region I and Region II with the same energy have different hybridization of $|{d}\rangle$ and $|{p}\rangle$, i.e. the overlapping between states in each regions is different. As the potential increases above $V=\Delta$ (Fig. 4(c)–(f)) a dome-like shape appears. The height of the blue dome, which corresponds to the Klein tunneling known so far, increases with $V$. On the other hand, the height of the orange dome does not change with $V$.

 

Fig. 4. Transmission coefficient $T$ as a function of $E$ at different $V$ values. The gray stripe shows the bandgap of the PhC in Region I. The insets depict the photonic bands at these $V$ values. The blue curve is the tunneling through the trivial-trivial interface with the Klein tunneling appearing at the large-$V$ regime $E\leq V-\Delta /2$. ’N’ and ’K’ stand for normal tunneling and Klein tunneling, respectively. The orange curve is the tunneling through the trivial-topological interface with the Klein tunneling appearing at the small-$V$ regime $E\geq V+\Delta /2$.

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4. Negative index of refraction

Negative index of refraction has been associated with the massive and massless Klein tunneling [9,14]. Here, we investigate whether this association is still valid for the trivial-topological interface. As in the previous sections we focus on transmission with wavevectors near the $\Gamma$ point. The physical velocity of a photonic quasiparticle is the group velocity which is defined as $\mathbf {v}_g=\hbar ^{-1}\mathrm {grad}_\mathbf {k}E$ [38], and is given from Eq. (15) and (16)

Component-wise we have $\mathbf {v}_g=(v_{g,x},v_{g,y})$. Here, we assume that $E$ lies outside of the bandgap such that Eq. (29) takes real values. By definition, $v_{g,x}$ is positive for the incident and transmitted states. This relation is preserved if $k_x^\mathrm {in}$ is proportional to $\mathrm {sgn}(E)$ and $k_x^\mathrm {t}$ is proportional to $\mathrm {sgn}(E-V)$, which has been depicted in Fig. 1(c) and (d). On the other hand, from continuity at the boundary we obtain $k^\mathrm {in}_y=k^\mathrm {t}_y$. If we choose $\phi ^\mathrm {in}$ and $E$ to be positive, then $k^\mathrm {in}_y$ and $k^\mathrm {t}_y$ are both positive. If $E-V>0$ then $v_{g,y}$ is positive for the incident and transmitted states, so the index of refraction is positive. On the other hand, if $E-V<0$, then $v_{g,y}$ is positive for the incident state but negative for the transmitted state, so the index of refraction is negative. This result can be generalized using Eq. (29) and we obtain

The above equation can be understood as an analog of Snell’s law where the index of refraction can be positive or negative depending on the values of $E$ and $V$ (Similar results are obtained in Refs. [9,14]), as shown in Fig. 5. In particular, we obtain a negative index of refraction in the large-$V$ regime which has $E>0$ and $E-V<0$, i.e. for tunneling from a concave-up band to a concave-down band (Fig. 5(c)). This situation is analogous to Ref. [38] which explains the negative refraction by a concave-down photonic band. Note that Eq. (30) is independent of the sign of $M^<$ and $M^>$ since the mass enters into the Hamiltonian as $M^2$. Therefore, negative refraction appears at the large-$V$ regime of both trivial-trivial and trivial-topological interfaces. On the other hand, we have shown in the previous section that for a trivial-topological interface, Klein tunneling appears in the small-$V$ regime while normal tunneling appears in the large-$V$ regime. This result implies that negative refraction is not directly related to massive Klein tunneling.

 

Fig. 5. (a) Dependence of the index of refraction on the potential $V$. The index of refraction is positive in the small-$V$ regime (green region) but negative in the large-$V$ regime (orange region). Note that this result is independent of the type of interface, i.e. independent of the mass sign. (b) and (c) Sign of the index of refraction at $V=0.5\Delta$ and $V=1.5\Delta$, respectively. Negative index of refraction appears for tunneling from a concave-up band to a concave-down band (or vice versa).

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5. Jackiw-Rebbi soliton

It is well known that the Jackiw-Rebbi soliton appears at the center of the bandgap of a positive-negative mass interface [35]. Here, we check whether such states reduce the transmission. As in previous studies [36] we split the Hamiltonian into

The wavefunction must vanish as $\lvert x\rvert \to \infty$ which requires $\kappa ^<$ and $\kappa ^>$ to be positive. After solving $\hat H_+(x)\psi _+(\mathbf {r})=E_0\psi _+(\mathbf {r})$ with Eq. (32) and Eq. (34) we obtain

These conditions are satisfied simultaneously if $E_0$ takes the following form

Therefore, the interfacial state is described by the following wavefunction

Figure 6 shows the energy of the interfacial state (black line) at different $V$ values. On top of that, we also plot the bulk band of the trivial PhC (blue curve) and topological PhC (orange curve) as a function of $k_y$ with fixed $k_x$ values. Since $\eta _s$, $\kappa ^\lessgtr$, $A^\lessgtr$ are positive parameters, the group velocity $v_{g,y}=\hbar ^{-1}\partial \Delta E/\partial k_y$ is negative, i.e. the soliton (with up spin) propagates in the negative $y$ direction. For the pseudospin-down sector, the soliton propagates in the positive-$y$ direction, which is a manifestation of pseudospin-momentum locking in topological interfaces with time reversal symmetry [26,39,40]. Explicitly, differentiating Eq. (44) with respect to $k_y$ and using Eq. (36), Eq. (39) and Eq. (41) we obtain

For $M^<=-M^>=\Delta /2$ and $A^>=A^<=A$ we obtain

 

Fig. 6. Plot of the bulk band of the trivial PhC (blue curve) and topological PhC (orange curve) as a function of $k_y$ with the interfacial state (black line). Here, we consider only the pseudospin-up states, so only one interfacial state appears due to pseudospin-momentum locking. (a)–(c): For $k_x>0$ the interfacial state is lower than the transmitted state (upper band of trivial PhC), so the interfacial state does not affect the transmission.

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Since $k_y$ is conserved, transition from the incident state to the Jackiw-Rebbi interfacial state is possible only if both states share the same $E$ and $k_y$ value. Figure 6(a)–(c) shows that the dispersion of the interfacial state is always lower than that of the transmitted state (upper band of trivial PhC). Therefore, we conclude that the Jackiw-Rebbi interfacial state does not affect the transmission.

6. Discussions

In this article, we point out that honeycomb-type PhCs provide an ideal platform to investigate the nature of Klein tunneling, where the effective Dirac mass can be tuned in a relatively easy way from a positive value (trivial PhC) to a negative value (topological PhC) via a zero-mass case (PhC graphene). We considered two types of interfaces, namely the trivial-trivial interface and the trivial-topological interface, where topological interfacial states does not affect transmission in the latter case. Our results can be verified experimentally by extending previous experimental methods on massless Klein tunneling in photonic crystals, such as Ref. [20]. The Dirac mass can be tuned by shifting the position of dielectric pillars/holes inward ($M>0$) or outward ($M<0$) from the center of the unit cell as in Refs. [26–31] and the potential V can be implemented by changing the effective permittivity of the PhC, which can be achieved by changing the average pillar/hole size.

Finally, we summarize our results and discuss how these results shed new light on the understanding of the Klein tunneling. First, by studying the transmission at both types of interfaces, we found that transmission at normal incidence at the trivial-trivial PhC interface with a large/small $V$ is identical to that of a trivial-topological interface with a small/large $V$. The reason for this duality is that in the large-V regime, the mass sign of the transmitted particle is effectively reversed at the trivial-trivial interface. While massive Klein tunneling has been defined as particle-to-antiparticle tunneling in the large-$V$ regime, we discover that, for the trivial-topological interface, particle-antiparticle tunneling occurs at the low-$V$ regime. Therefore, we conclude that the high potential is not necessary for the definition of Klein tunneling. Second, we considered the way of refraction at PhC interfaces and found that transmission with a negative index of refraction is achieved in the large-$V$ regime both for the trivial-trivial and trivial-topological interfaces. In fact, it has been shown that negative refraction can be achieved with a photonic band with concave-down curvature [38], which is what we obtain in the large-$V$ regime considered by Klein. While negative index of refraction, which occurs at the large-$V$ regime, has been associated with massive and massless Klein tunneling, here we have shown that massive Klein tunneling appears in the small-$V$ regime of a trivial-topological interface. Therefore, the large potential and massive Klein tunneling should be considered separately. Our results are not limited to PhC systems but also apply to other Dirac systems.