Fermi surface anisotropy in plasmonic metals increases the potential for efficient hot carrier extraction

Realizing the potential of plasmonic hot carrier harvesting for energy conversion and photodetection requires new materials that resolve the bottleneck of extracting carriers prior to energy relaxation within the metal. Using first-principles calculations of optical response and carrier transport properties, we show that directional conductors with Fermi velocities restricted predominantly to one or two directions present significant advantages for efficient hot carrier harvesting. We show that the optical response of filmlike conductors ${\mathrm{PtCoO}}_2$ and ${\mathrm{Cr}}_2\mathrm{Al}\mathrm{C}$ resembles that of two-dimensional (2D) metals, while that of wirelike conductors CoSn and $\mathrm{Y}{\mathrm{Co}}_3{\mathrm{B}}_2$ resembles that of 1D metals, which can lead to high mode confinement and efficient light collection in small dimensions, while still working with 3D materials with high carrier densities. Carrier lifetimes and transport distances in these materials, especially in ${\mathrm{PtCoO}}_2$ and CoSn, are competitive with noble metals. Most importantly, we predict that the carrier injection efficiency from all of these materials into semiconductors can exceed 10% due to the small component of carrier momentum parallel to the metal surface, substantially improving upon the typical injection efficiency of less than 0.1% from noble metals into semiconductors.

Figure 1

Unit cell structure and Fermi surfaces of the directional conductors studied here. Note that all are 3D hexagonal crystals, but have Fermi velocities (Fermi surface normals) along the $a$ directions for the filmlike conductors [(a) and (b)] and along the $c$ direction for the wirelike conductors [(c) and (d)]. The electron-phonon mean free path $\lambda$ varies substantially over the Fermi surface for all four materials.

Figure 2

(a) Transport-weighted Eliashberg spectral function and (b) frequency-dependent Drude relaxation time for directional conductors. ${\mathrm{PtCoO}}_2$ exhibits the lowest electron-phonon coupling strength [lowest ${\alpha }^{2}F\left(\omega \right)]$ and hence the highest relaxation time ${\tau }_D$, while ${\mathrm{Cr}}_2\mathrm{Al}\mathrm{C}$ has the strongest coupling and lowest ${\tau }_D$. The relaxation time drops from the $\omega =0$ value for $\omega \sim {\omega }_{\text{ph}}$ and saturates to a high-frequency limit for $\omega \gg {\omega }_{\text{ph}}$ for each material.

Figure 3

Complex dielectric function for the directional conductors along (a) the preferred transport direction $\widehat{\mathbf{j}}$ and (b) the corresponding normal direction $\widehat{\mathbf{n}}$, with real and imaginary parts in the left and right columns, respectively. The plasmonic window ($Re\epsilon <0$) is much wider along the $\widehat{\mathbf{j}}$ direction than along the $\widehat{\mathbf{n}}$ direction in all cases, making the optical response of the film conductors ${\mathrm{PtCoO}}_2$ and ${\mathrm{Cr}}_2\mathrm{Al}\mathrm{C}$ similar to 2D materials and that of wire conductors CoSn and $\mathrm{Y}{\mathrm{Co}}_3{\mathrm{B}}_2$ similar to that of 1D materials, even though all four materials are 3D materials with strong bonding in all directions.

Figure 4

(a)–(d) Hot carrier distribution, $P(\omega ,\varepsilon )$, as a function of carrier energy $\varepsilon$ and plasmon frequency $\omega$ for directional conductors, with plasmon electric field polarized along their corresponding $\widehat{\mathbf{j}}$ directions (left panels) and $\widehat{\mathbf{n}}$ directions (right panels). The probability is normalized such that $P(\omega ,\varepsilon )=1$ for a uniform electron and hole energy distribution [28]. Note that direct transitions dominate at almost all plasmon frequencies, except for $\hslash \omega \ll 1$ eV in ${\mathrm{PtCoO}}_2$ and CoSn, leading to carrier distributions that are sensitive to electronic structure details.

Figure 5

Fraction of interband absorption (left column) and fraction of energy in hot electrons (right column) for the directional conductors for light polarized along (a) the preferred transport direction $\widehat{\mathbf{j}}$ and (b) the corresponding normal direction $\widehat{\mathbf{n}}$. Interband absorption is stronger along the $\widehat{\mathbf{j}}$ direction but is dominated by direct absorption for photons with energy larger than 1 eV in all cases. Energy is evenly distributed between electrons and holes, except for low-energy intraband-dominated absorption of $\widehat{\mathbf{n}}$-polarized light in ${\mathrm{PtCoO}}_2$.

Figure 6

(a)–(d) Lifetime of hot carriers as a function of carrier energy in directional conductors, with color indicating relative contributions of electron-phonon and electron-electron scattering. Carrier lifetimes are comparable in magnitude to noble metals, especially in ${\mathrm{PtCoO}}_2$ and CoSn, but drop off with energy faster than for noble metals [28].

Figure 7

(a)–(d) Mean free path of hot carriers as a function of carrier energy in directional conductors, with behavior closely following those of the lifetimes shown in Fig. 6.

Figure 8

(a) Average velocity direction $\widehat{\mathbf{v}}$ component along preferred transport direction $\widehat{\mathbf{j}}$ in directional conductors compared with gold, as a function of carrier energy. (The dotted line at $\langle |\widehat{\mathbf{v}}·\widehat{\mathbf{j}}|\rangle =1/2$ indicates the isotropic limit.) The preferred orientation of velocities substantially increases the probability of carrier injection into a semiconductor (assumed parabolic bands with $m^{*}=0.3m_e$) with Schottky barrier heights (b) 0.2 eV and (c) 0.5 eV. Note the switch from logarithmic to linear scaling in the $y$ axis of (b) and (c) to simultaneously show the small injection probabilities from gold and the much larger injection probabilities from all directional conductors.