Current-voltage characteristics of Weyl semimetal semiconducting devices, Veselago lenses, and hyperbolic Dirac phase

The current-voltage characteristics of a new range of devices built around Weyl semimetals has been predicted using the Landauer formalism. The potential step and barrier have been reconsidered for three-dimensional Weyl semimetals, with analogies to the two-dimensional material graphene and to optics. With the use of our results we also show how a Veselago lens can be made from Weyl semimetals, e.g., from NbAs and NbP. Such a lens may have many practical applications and can be used as a probing tip in a scanning tunneling microscope (STM). The ballistic character of Weyl fermion transport inside the semimetal tip, combined with the ideal focusing of the Weyl fermions (by Veselago lens) on the surface of the tip may create a very narrow electron beam from the tip to the surface of the studied material. With a Weyl semimetal probing tip the resolution of the present STMs can be improved significantly, and one may image not only individual atoms but also individual electron orbitals or chemical bonding and therewith to resolve the long-term issue of chemical and hydrogen bond formation. We show that applying a pressure to the Weyl semimental, having no center of spatial inversion, one may model matter at extreme conditions, such as those arising in the vicinity of a black hole. As the materials ${\mathrm{Cd}}_3{\mathrm{As}}_2$ and ${\mathrm{Na}}_3\mathrm{Bi}$ show an asymmetry in their Dirac cones, a scaling factor was used to model this asymmetry. The scaling factor created additional regions of no propagation and condensed the appearance of resonances. We argue that under an external pressure there may arise a topological phase transition in Weyl semimetals, where the electron transport changes character and becomes anisotropic. There a hyperbolic Dirac phase occurs where there is a strong light absorption and photocurrent generation.

Figure 1

Diagram of the potential step problem. The two independent regions of different Weyl semimetal materials have been labled as $a$ and $b$. This is a simple model of a heterostructure made of two Weyl semimetals, e.g., the semimetal $a$ can be NbAs while the semimetal $b$ can be NbP. An external potential $V_a$ separating them is associated with affinity or a work function of these two different materials. A potential step is placed in the $x$ direction with a height $V_a\ne 0$ and $V_b=0$. The shaded region shows where hole transport is present.

Figure 2

Density plots for transmission against energy and incident angle for a potential step. The step shown has the heights $V_a=0$ eV and $V_b=0.1$ eV. (Top) The energy dependence is then shown with ${\varphi }_a=\pi /2$. (Bottom) The angular dependence is shown in the step for an energy of 0.05 eV.

Figure 3

Diagram of the potential barrier problem. It can be made from two or three different Weyl semimetals by combining two heterostructures similar to the ones demonstrated in Fig. 1. A potential barrier is placed in the $x$ direction with a height $V_b$ and a width $d$. The shaded region shows where hole transport is present. The three independent regions have been labeled as $a,b$, and $c$ and may correspond, for example, to Weyl semimetals such as NbAs, NbP, and TaAs. Such a three-layer Weyl semimetal structure can also act as a Veselago lens.

Figure 4

Density plots for transmission against energy and incident angle with one fixed angle. The potential barrier has a height $V_b=0.1$ eV and width $d=100$ nm. (Top) The energy dependence is then shown with ${\varphi }_a=\pi /2$. (Bottom) The angular dependence is shown in the step for an energy of 0.05 eV.

Figure 5

A schematic diagram showing a deformation of the electronic spectrum of Weyl semimetal under pressure P; the tilting angle of the Dirac cone increases and it finally takes a horizontal position, i.e., tips over. Such tilting of the cone corresponds also to the dramatic changes in the shape and topology of Fermi surface. If at a small angle the Fermi surface has a closed shape of a deformed circle or sphere then at the tipped stage, the Fermi surface is open and consists of two separate parts, see the dark blue areas on the insert in the center of the figure. The character of the electronic transport undergoes a dramatic transformation from ballistic and isotropic to diffusive and very anisotropic. Such a transformation of the electronic spectrum in tilted Weyl semimetals strongly resembles the behavior of a light cone when a probe particle is approaching a black hole horizon, where space and time coordinates are interchanged [55].

Figure 6

Density plots for transmission probability against energy and incident angle from Eq. (18) with the modified $k_z$ from Eq. (23). The potential barrier shown has the characteristics $V_a=0$ eV, $V_b=0.1$ eV, and $\lambda =1/3$. (Top) The energy dependence is then shown with ${\varphi }_a=\pi /2$. (Bottom) The angular dependence is shown in the step for an energy of 0.05 eV.

Figure 7

A schematic operation of a scanning tunneling microscope (STM) with probing tip built from Weyl semimetals (given in green and orange). It is made in the form of a Veselago lens, e.g., in the form of three layers of NbAs and NbP as given in the main pane. In this case, for the electron propagation through, these three layers act as a rectangular potential barrier (shown in blue on the right side). The electron flow supplied by the electrical current to the top layer is focused on a small spot in the bottom layer (given in red). Due to the properties of the Veselago lens associated with the materials negative refractive index, the focusing spot for electrons can be very small and may increase the spatial and temporal resolution of the STM. By measuring the current-voltage characteristics $I\left(V\right)$ and scanning the surface one may, in principle, identify various electron clouds, atomic orbitals, and resolve the existing issues with chemical bonding and kinetics.

Figure 8

A simple example of a Weyl semimetal transistor with current-voltage characteristics described by Eq. (38).

Figure 9

The gate voltage ($V_g$) dependence on current for a Weyl semimetal transistor from Eq. (38), with $I_0=e\frac{2L_y{L}_z}{\pi {\hslash }^3{v}_f^2}$, $V_b=V_g$, $V_{sd}=0.2$ eV, $d=100$ nm, and $T=298$ K.

Figure 10

The source-drain voltage ($V_{\text{sd}}$) dependence on current for a Weyl semimetal transistor from Eq. (38) with $I_0=e\frac{2L_y{L}_z}{\pi {\hslash }^3{v}_f^2}$, $V_b=0.1$ eV, $d=100$ nm, and $T=298$ K.

Figure 11

The temperature dependence on current for a Weyl semimetal transistor from Eq. (38) with $I_0=e\frac{2L_y{L}_z}{\pi {\hslash }^3{v}_f^2}$, $V_b=0.1$ eV, $d=100$ nm, and $V_{\text{sd}}=0.2$ eV.