Atomic collapse in pseudospin-1 systems

We investigate atomic collapse in pseudospin-1 Dirac material systems whose energy band structure constitutes a pair of Dirac cones and a flat band through the conic intersecting point. We obtain analytic solutions of the Dirac-Weyl equation for the three-component spinor in the presence of a Coulomb impurity and derive a general criterion for the occurrence of atomic collapse in terms of the normalized strength of Coulomb interaction and the angular momentum quantum number. In particular, for the lowest angular momentum state, the solution coincides with that for pseudospin-1/2 systems, but with a reduction in the density of resonance peaks. For higher angular momentum states, the underlying pseudospin-1 wave functions exhibit a singularity at the point of zero kinetic energy. Divergence of the local density of states associated with the flat band leads to an inverse square type of singularity in the conductivity. These results provide insights into the physics of the two-body problem for relativistic quantum pseudospin-1 quasiparticle systems.

Figure 1

Pseudospin-1 lattice and energy band structure. (a) A pseudospin-1 lattice where the three sublattices of nonequivalent atoms are distinguished by color (red, blue, and green, respectively). A Coulomb impurity is shown as the red cross. (b) Energy band structure in the presence of the impurity scattering atom. At each point, the energy band constitutes a pair of Dirac cones and a flat band through the conic intersecting point. Because of the Coulomb potential, locally the height of the three-band structure depends on the position. The black and blue dashed lines indicate the energy $E_i$ of the incident particle and the kinetic energy $E-V$, respectively.

Figure 2

Possible atomic collapse states for pseudospin-1 and pseudospin-1/2 systems. As the strength $\left|g\right|$ of the Coulomb impurity is increased, more and more collapse states are possible, each associated with a distinct angular momentum number. For pseudospin-1 systems, the second spinor component can be conveniently used to find the collapse states. For pseudospin-1/2 systems, the condition under which atomic collapse occurs is that the quantity $s=\sqrt{l^2-g^2}$ becomes imaginary, where $l$ is the angular momentum quantum number associated with the first spinor component plus $1/2$. Atomic collapse states can arise but only for $\left|g\right|>1/2$. For example, for $1/2<\left|g\right|\le 1$, the shaded region indicates that there is only one collapse state for a pseudospin-1 system, which occurs at $l=0$, but for a pseudospin-1/2 system, two such states exist: $l=\pm 1/2$.

Figure 3

Scattering wave function in the collapse-free regime. For $g=1/5$, (a) spinor component ${\psi }_A$, where the vertical dashed line denotes $r=0$. As indicated by Eq. (6), in this regime the Coulomb impurity will generate a singularity. (b), (c) Spinor components ${\psi }_B$ and ${\psi }_C$, respectively. The singularity at $\rho =-g$ is a result of the flat band. In the presence of an impurity, the solutions associated with the flat band and with the Dirac cones are mixed. In terms of the angular momentum, the singularity occurs for $l\ne 0$. (d)–(f) The absolute values of ${\psi }_A, {\psi }_B$ and ${\psi }_C$, respectively for $\left|\rho \right|\gg 1$, where each color dashed line indicates the wave-function component for negative values of $\rho$. The dashed black envelope line is indicative of the decay behavior $\sqrt{2/\left(\pi r\right)}$ given by Eq. (11).

Figure 4

Local density of states (LDOS) of pseudospin-1 systems in the collapse-free regime. Shown are patterns of LDOS at the location $r=a$, for different values of $g$. The dashed lines indicate the free space DOS where $N(E,r)=\left|E\right|/\left(2\pi \right)$ and $N\left(0\right)=\infty$.

Figure 5

Local density of states (LDOS) of a pseudospin-1 system in the collapse-prone regime. For $g=1$ and truncation at $a_0=0.5a$ (a) LDOS for different values of $r$ where the rapid oscillations near $E\to 0^{-}$ are indicative of atomic collapse, (b1) LDOS for the zero angular momentum ($l=0$) mode on a logarithmic energy scale and (b2) LDOS for the $l=\pm 1/2$ modes for the corresponding pseudospin-1/2 system. (c) Oscillatory behavior in the LDOS difference $\delta N=N_{g=1}-N_{g=0}$—a signature of atomic collapse.

Figure 6

Flat-band induced divergence of conductivity in pseudospin-1 systems. (a) Shown is the behavior of LDOS near $E_0$ for $g=1/5$ and $r=a$, where $E_0=-3t/\left(5\sqrt{2}\right)\approx -0.42t$ [Eq. (15)]. There is a singularity in the spinor wave function determined by $\rho +g=0$ [or, equivalently, $E-V\left(r\right)=0]$. In this case, the LDOS and hence the conductivity exhibits a ${(E-E_0)}^{-2}$ type of divergence as the energy approaches $E_0$. (b) The same quantity for $g=1$.

Figure 7

Comparison between pseudospin-1/2 and pseudospin-1 systems. The parameters are $E=-t$ (constant bias) and $g=1/5$. (a) Conductivity for a pseudospin-1/2 system, where space is in units of the atomic lattice spacing $a$. The values of the conductivity are color coded on a logarithmic scale with $1/\left(2\pi \right)$ being the free space conductivity. (b) Conductivity for a pseudospin-1 system, where the legends are the same as in (a). (c) Current in the pseudospin-1/2 system associated with the $l=1/2$ state, where $j_r=0$ and $j_{\theta }$ is continuous when passing through the zero kinetic energy region (dashed line). (d) Current in the pseudospin-1 system associated with the $l=1$ state, where $j_{\theta }$ changes sign when passing through the zero kinetic energy region at which the value of the current diverges.