Collective excitations in three-dimensional Dirac systems

We provide the plasmon spectrum and related properties of the three-dimensional (3D) Dirac semimetals ${\mathrm{Na}}_3\mathrm{Bi}$ and ${\mathrm{Cd}}_3{\mathrm{As}}_2$ based on the random-phase approximation. The necessary one-electron eigenvalues and eigenfunctions are obtained from an effective $\mathbf{k}·\mathbf{p}$ Hamiltonian. Below the energy at which the velocity $v_z$ along the $k_z$ axis vanishes, the density of states differs drastically from that of a 3D electron gas (3DEG) or graphene. The dispersion relation is anisotropic for wave vectors parallel ($q$) and perpendicular ($q_z$) to the $(x,y)$ plane and is markedly different than that of graphene or a 3DEG. The same holds for the energy-loss function. Both depend sensitively on the position of the Fermi energy $E_F$ relative to the region of the Berry curvature of the bands. For $E_F$ below the energy at which $v_z$ vanishes, the range of the relevant wave vectors $q$ and $q_z$ shrinks, for $q_z$ by about one order of magnitude.

Figure 1

(a) Energy dispersion $E_{\mathbf{K}}$ of ${\mathrm{Na}}_3\mathrm{Bi}$ as a function of $k$ and $k_z$, as given by Eq. (3). (b) $E_{\mathbf{K}}$ as a function of $k$ at $k_z=k_c$. The Dirac point is at $k=0$ with energy $E_0=7.6$ meV. (c) $E_{\mathbf{K}}$ as a function of $k_z$ at $k=0$. There are two Dirac points at $k_z=\pm k_c$. The top of the Berry curvature in the conduction band is $E_1\simeq 23$ meV, its bottom in the valence band at $E_2\simeq -151$ meV, and they are both located at $k=0$ and $k_z=0$. (d)–(f) As in (a)–(c), respectively, for ${\mathrm{Cd}}_3{\mathrm{As}}_2$. The energy of the Dirac point is at $E_0\simeq -218.68$ meV. The top of the Berry curvature in the conduction band is $E_1\simeq -209$ meV and its bottom in the valence band at $E_2\simeq -229$ meV. The red curve shows the conduction band and the blue-dash-dotted curve is the valance band. The green-dashed and black-dotted lines show the Fermi energy $E_F$ for high- and low-electron densities, respectively, see Figs. 3, 4, 5, 6.

Figure 2

Density of states (DOS) $D\left(E\right)$ for ${\mathrm{Na}}_3\mathrm{Bi}$ as a function of the electron energy $E$ (blue curve) with $D_0=C_0^2/A^3$ for ${\mathrm{Na}}_3\mathrm{Bi}$. The top of the Berry curvature in the conduction band is $E_1\simeq 23.2$ meV and is marked by the red-dashed curve. At this value of the energy, the velocity $\partial E_K/\partial k_z$ vanishes (cf. Fig. 1), i.e., it corresponds to an integrable van Hove singularity. The DOS for a 3DEG is shown by the orange-dotted curve using the effective mass $m^{*}\simeq 0.11m_e$ of ${\mathrm{Na}}_3\mathrm{Bi}$ and $D_0=m^{*3/2}\sqrt{2|C_0|}/\left(\pi {\hslash }^3\right)$ for a 3DEG. The black-dashed curve is the result for graphene with energy dispersion $E=\hslash v_{F}k$ and $D_0=C_0^2/{\left(\hslash v_F\right)}^3$. Arrows indicate the two values of $E_F$ used to obtain the results shown in Figs. 3, 4, 5, 6.

Figure 3

Dispersion relations and energy loss functions in ${\mathrm{Na}}_3\mathrm{Bi}$ along different $\mathbf{Q}$ directions at temperature $T=10$ K, electron density $N_e=1\times {10}^{19}{\mathrm{cm}}^{-3}$ [see green-dashed lines in Figs. 1 and 1, corresponding to $E_{F1}\simeq 59.076$ meV], and lifetime $\tau =6.71$ ps. In (a) and (c), we have $Q_{\parallel }=(\mathrm{q},\phi ,0)$ at $\phi =0$; the Fermi wave vector $k_F$ along the $k$ direction is about $\sim 2.25\times {10}^8{\mathrm{m}}^{-1}$. In (b) and (d), we have $Q_{\perp }=(0,0,q_z)$; the Fermi wave vector $k_{Fz}$ along the $k_z$ direction is about $\sim 10.4\times {10}^8{\mathrm{m}}^{-1}$. The orange-dotted curve in (a) is graphene's plasmon dispersion relation for $N_e=1\times {10}^{12}{\mathrm{cm}}^{-2}$. The black-dash-dotted curves in (a) and (b) represent, for a 3DEG with $m^{*}\simeq 0.24m_e$, the beginning of the particle-hole (p-h) excitations area in which $\omega \simeq {\hslash }^2(q^2+2q{k}_F)/2m^{*}$.

Figure 4

As in Fig. 3 with electron density $N_e=1\times {10}^{17}{\mathrm{cm}}^{-3}$ [see black-dotted lines in Figs. 1 and 1, corresponding to $E_{F2}\simeq 14.634$ meV]. For $Q_{\parallel }=(\mathrm{q},\phi ,0)$ at $\phi =0$, the Fermi wave vector $k_F$ along the $k$ direction is about $\sim 0.298\times {10}^8{\mathrm{m}}^{-1}$. For $Q_{\perp }=(0,0,q_z)$, the Fermi wave vector $k_{Fz}$ along the $k_z$ axis is about $\sim 9.24\times {10}^8{\mathrm{m}}^{-1}$.

Figure 5

Plasmon dispersions and energy loss functions in ${\mathrm{Cd}}_3{\mathrm{As}}_2$ along different $\mathbf{Q}$ directions at fixed temperature $T=10$ K, electron density $N_e=1\times {10}^{19}{\mathrm{cm}}^{-3}$ [see Figs. 1 and 1, corresponding to $E_{F1}\simeq -185.651$ meV], and lifetime $\tau =6.87$ ps. For $Q_{\parallel }=(\mathrm{q},\phi ,0)$ at $\phi =0$, the Fermi wave vector $k_F$ along the $k$ direction is about $\sim 1.33\times {10}^8{\mathrm{m}}^{-1}$. For $Q_{\perp }=(0,0,q_z)$, the Fermi wave vector $k_{Fz}$ along the $k_z$ axis is about $\sim 0.68\times {10}^8{\mathrm{m}}^{-1}$. The orange-dotted curve in (a) is graphene's dispersion relation for $N_e=1\times {10}^{12}{\mathrm{cm}}^{-2}$. The black-dash-dotted curves in (a) and (b) represent, for a 3DEG with $m^{*}\simeq 0.24m_e$, the beginning of the particle-hole (p-h) excitations area in which $\omega \simeq {\hslash }^2(q^2+2q{k}_F)/2m^{*}$.

Figure 6

As in Fig. 5 with electron density $N_e=1\times {10}^{17}{\mathrm{cm}}^{-3}$ [see Figs. 1 and 1, corresponding to $E_{F2}\simeq -211.237$ meV]. For $Q_{\parallel }=(\mathbf{q},\phi ,0)$ at $\phi =0$, the Fermi wave vector $k_F$ along the $k$ direction is about $\sim 0.301\times {10}^8{\mathrm{m}}^{-1}$). For $Q_{\perp }=(0,0,q_z)$, the Fermi wave-vector $k_{Fz}$ along the $k_z$ axis is about $\sim 0.438\times {10}^8{\mathrm{m}}^{-1}$.

Figure 7

(a) Plasmon dispersion for ${\mathrm{Na}}_3\mathrm{Bi}$ as a function of $q$ at fixed temperature $T=10$ K, electron density $N_e=1\times {10}^{19}{\mathrm{cm}}^{-3}$, lifetime $\tau =6.71$ ps, and fixed $q_z$. (b) Plasmon dispersion for ${\mathrm{Na}}_3\mathrm{Bi}$ with $q$ and $q_z$ interchanged is shown. (c), (d) As in (a) and (b), respectively, for ${\mathrm{Cd}}_3{\mathrm{As}}_2$, for the same temperature and electron density, and $\tau =6.87$ ps. All curves for selected $q_z$ or $q$ isovalues are marked as shown in the insets.