Elucidating Na$_2$KSb band structure: near-band-gap photoemission spectroscopy and DFT calculations

TL;DR

The paper integrates near-band-gap photoemission spectroscopy at $T=80$ K with SOC-enabled DFT calculations to map the band structure of Na$_{2}$KSb(Cs,Sb) photocathodes. It identifies $E_{ ext{g}} = 1.52 \pm 0.02$ eV, $Δ_{ ext{SO}} = 0.59 \pm 0.04$ eV, and CB side-valley offsets $Δ_{\Gamma-\text{X}1} = 0.41 \pm 0.05$ eV and $Δ_{\Gamma-\text{X}2} = 0.65 \pm 0.05$ eV, corroborated by DFT-1/2 calculations. The analysis of EDC derivatives reveals ballistic and hot-electron emission channels from HH, LH, and SO valence bands and confirms intervalley scattering to X valleys, consistent with the calculated CB topology. The findings show good agreement between experiment and theory, including non-parabolicity and anisotropy near Γ, and have implications for optimizing spin-polarized electron sources based on Na$_{2}$KSb(Cs,Sb).

Abstract

The electronic band structure of Na$_{2}$KSb was studied by a combination of low-energy photoemission spectroscopy and density functional theory (DFT) calculations. The optical and photoemission quantum efficiency (QE) spectra, along with longitudinal energy distribution curves (EDCs) of multialkali Na$_{2}$KSb(Cs,Sb) photocathodes were measured in the temperature range of 80--295 K. The thresholds of various band-to-band transition in Na$_{2}$KSb were observed in the optical and QE spectra of Na$_{2}$KSb(Cs,Sb) photocathodes. The evolution of EDC derivatives with varying photon energy reveals a fine structure related to the emission of two types of electrons: (i) ballistic electrons, which are excited from heavy hole, light hole and split-off valence bands, and (ii) photoelectrons, that are captured in the side valleys of Na$_{2}$KSb conduction band. The analysis of EDCs and QE spectra allowed us to determine the band structure parameters of Na$_{2}$KSb at $T = 80$ K, including the band gap $E_{\text{g}} = 1.52 \pm 0.02$ eV, spin-orbit splitting $Δ_{\text{SO}} = 0.59 \pm 0.04$ eV and the energy separations between $Γ$ and side valleys of the conduction band: $Δ_{Γ-\text{X}1} = 0.41 \pm 0.05$ eV and $Δ_{Γ-\text{X}2} = 0.65 \pm 0.05$ eV. The experimentally determined band gaps and side valley positions, as well as the energies of the final electronic states of optical transitions are in good agreement with the DFT calculations. The obtained data on the hot electron dynamics and electronic band structure of Na$_{2}$KSb are crucial to improve the understanding of the photoemission processes in this material and will contribute to the development of the robust spin-polarized electron sources with multialkali photocathodes.

Figures (5)

• Figure 1: (a) Schematic energy band diagram of the Na$_{2}$KSb(Cs,Sb) photocathode with NEA and the photoemission pathways for ballistic, hot and thermalized electrons. $E_{\text{c}}$ and $E_{\text{v}}$ are the conduction band bottom and the valence band top in Na$_{2}$KSb, respectively, $E_{\text{g}}$ is the band gap, $E_{\text{vac}}$ is the vacuum level, $\chi^{*} = E_{\text{c}} - E_{\text{vac}}$ is the effective electron affinity, $E_{\text{F}}$ is the Fermi level, $E_{\text{0}}$ is the energy of ballistic electrons, $E_{\text{c}\text{X}}$ is the bottom of the conduction band side valley, $m_{\text{e}}$ and $m_{\text{h}}$ are the effective masses of electrons and holes, respectively. (b) The illustration of the dependence of characteristic electron energies on the excitation photon energy $\hbar\omega$.
• Figure 2: (a) Schematic cross-section and (b) photograph of a vacuum photodiode with the Na$_{2}$KSb(Cs,Sb) photocathode and a semitransparent anode. (c) Photoemission quantum efficiency spectra of the Na$_{2}$KSb(Cs,Sb) photocathode measured in transmission (T-mode) and reflection (R-mode) illumination geometries at $T = 80$ K. The threshold energies for transitions from the valence bands to the conduction band ($E_{\text{g}}$ and $E_{\text{g}}+\Delta_{\text{SO}}$) are marked with arrows. The R-mode QE spectrum is normalized by the anode transmission spectrum. (d) Typical EDCs and (e) their derivatives of the Na$_{2}$KSb(Cs,Sb) photocathode measured in the R-mode at $T = 80$ K. The hot and thermalized electron emission components are indicated for the EDC measured at $\hbar\omega =$2.45 eV. The vacuum level $E_{\text{vac}}$, conduction band bottom in the bulk of Na$_{2}$KSb $E_{\text{c}}$ and the initial kinetic energy of the ballistic electrons in the conduction band $E_{\text{0}}$ are marked with arrows for EDC and its derivative measured at $\hbar\omega = 1.85$ eV.
• Figure 3: The evolution of the EDC derivatives of the Na$_{2}$KSb(Cs,Sb) photocathode with varying $\hbar\omega$ measured in the R-mode at $T = 80$ K. The derivatives are normalized to their maximum absolute values in the hot EDC region of $E_{\text{lon}} > 0.3$ eV (the curves measured at $\hbar\omega = 1.55$ eV and 1.65 eV were normalized by the same value) and shifted vertically for clarity. Negative peaks corresponding to emission of ballistic electrons and hot electrons, scattered to the upper valleys of the conduction band, are marked for $\hbar\omega = 2.45$ eV and $\hbar\omega = 3.25$ eV: HH-$\Gamma_{\text{CB}}$(square), LH-$\Gamma_{\text{CB}}$(circle), SO-$\Gamma_{\text{CB}}$(diamond), $\text{X}_{\text{CB1}}$ (triangle) and $\text{X}_{\text{CB2}}$ (open triangle). The evolutions of peak positions with varying $\hbar\omega$ are highlighted with dashed lines as a guides to the eye. The position of the conduction band minima $E_{\text{c}}$ is marked with an arrow.
• Figure 4: The evolution of the peak energy positions in EDC derivatives with varying $\hbar\omega$ for the Na$_{2}$KSb(Cs,Sb) photocathode measured at $T = 80$ K. The DFT-calculated energies of ballistic electrons along the $\Gamma$-X direction and the bottoms of the upper valleys of the conduction band are shown with solid lines. The calculated curves are shifted toward the higher $\hbar\omega$ by 0.11 eV to account for the difference between the calculated and measured band gaps (1.41 eV and ($1.52 \pm 0.02$) eV, respectively). The experimentally determined threshold energies of optical transitions from valence bands to the conduction band ($E_{\text{g}}$ and $E_{\text{g}}+\Delta_{\text{SO}}$) and the positions of the conduction band minima ($E_{\text{c}}$, $E_{\text{c}\text{X1}}$ and $E_{\text{c}\text{X2}}$) are marked with arrows.
• Figure 5: (a) Balls-and-sticks representation of the primitive cell of cubic Na$_{2}$KSb. (b) The Brillouin zone of Na$_{2}$KSb. The high-symmetry points and the path connecting them are highlighted. (c) The calculated band structure of Na$_{2}$KSb. $E_{\text{c}}$ and $E_{\text{v}}$ are conduction band bottom and valence band top, respectively. The energy gaps ($E_{\text{g}}$, $\Delta_{\text{SO}}$, $\Delta_{\Gamma-\text{X}1}$, $\Delta_{\Gamma-\text{X}2}$), are highlighted. (d) The calculated band structure of Na$_{2}$KSb at the center of the Brillouin zone along $\Gamma$-X, $\Gamma$-K and $\Gamma$-L directions, illustrating the bands non-parabalicity and anisotropy. The parabolic band approximations for the conduction band (CB), heavy hole (HH), light hole (LH) and split-off (SO) valence bands are shown by dashed lines for the $\Gamma$-X direction along with respective effective masses (in units of the free electron mass).