Generalizable density functional theory based photoemission model for the accelerated development of photocathodes and other photoemissive devices

In this work, we have developed an ab initio photoemission model that accurately describes the photoemission process for the most diverse range of photocathode materials to date. Compared to previous photoemission models, this is accomplished by considerably reducing the number of approximations and assumptions used in representing the photoemission process and the photoemitting material itself. Notably, our model directly includes the full electronic structure of the material, photoexcitation probabilities for all direct optical transitions, and an improved surface-vacuum barrier transmission probability. To test the performance of our model, we perform validations with experimental measurements for all photocathode materials studied in this work. Whereas previous models have often qualitatively disagreed with the measured photoemission properties of some materials, our model is found to provide quantitative agreement with experimental measurements for all tested materials. As an example, our method predicts the root-mean-square transverse momentum of electrons emitted from PbTe up to an excess energy of 1.0 eV with a mean absolute error that is ∼5× less than from previously derived expressions. Perhaps more importantly, our model is able to match experimentally observed decreases in intrinsic emittance with increasing photon energy—a feat that current analytical models are unable to achieve. We expect that the broad applicability of our model will greatly accelerate the rate of discovery, characterization, and scientific understanding of photocathodes and other photonic devices.

Figure 1

Workflow for calculating the intrinsic emittance for a given material. We first obtain the relaxed crystal structure of the bulk material. We then calculate the electronic structure of the material with KS-DFT. Finally, we determine the total escape probability ($w_i$) associated with all Kohn-Sham states and calculate the intrinsic emittance,${\varepsilon }_{\mathrm{int}}$.

Figure 2

Calculated PbTe electronic band structure. For an incident photon of energy, $\hslash \omega$, only electrons excited into the conduction-band states with energy above the vacuum level, $E_{\mathrm{vacuum}}=E_{\mathrm{Fermi}}+\varphi$, have sufficient energy to escape the photocathode surface-vacuum interface. The maximum energy a conduction-band state can have to undergo photoexcitation is $\hslash \omega$, relative to the Fermi level. As such, the relevant conduction-band states have energy $E_c$ between $E_{\mathrm{vacuum}}\le E_c\le \hslash \omega$. The minimum energy that a valence-band state must have to undergo photoexcitation followed by emission to vacuum is given by $E_{\mathrm{vacuum}}-\hslash \omega$.The above plot of the PbTe electronic band structure shows all Kohn-Sham states that may contribute to photoemission with a photon energy of 9 eV and a vacuum level of 4 eV, relative to the Fermi level.

Figure 3

Weights used in the calculation of the PbTe (111) intrinsic emittance: (a) Dirac delta function weight given by Eq. (17) at $\hslash \omega =5\mathrm{eV}$, (b) the photoexcitation matrix elements, $\langle {\psi }_{\vec{k},c}|\vec{p}|{\psi }_{\vec{k},v}{\rangle }^{2}F\left(T\right)$, seen in Eq. (13), (c) the electron escape probability, $w_{i,\mathrm{escape}}$, given by Eq. (30), and (d) the total electron emission weight $w_i$ given by Eq. (8) at $\hslash \omega =5\mathrm{eV}$. The horizontal red lines in (a) and (d) correspond to the incident photon energy, $\hslash \omega =5\mathrm{eV}$, whereas the red overlaid plot in (d) gives $\sqrt{\langle k_x^2\rangle }$ as a function of $\hslash \omega$. The conduction-band states below the vacuum level are omitted here for clarity. The plots shown here are for the PBE calculated PbTe states without fitting to the experimental band gap or electron effective mass. The weights in (d) are calculated as the product of the weights in plots (a)–(c). A video illustrating the incident photon energy dependence of (d) is given in the Supplemental Material [22].

Figure 4

Bulk PbTe(111) intrinsic emittance at 300 K calculated with various different total emission probabilities $w_i$. The orange line exhibits a nonmonotonic relationship between intrinsic emittance and energy, indicating that the delta function,$\delta \left[E_c\left(\vec{k}\right)-E_v\left(\vec{k}\right)-\hslash \omega \right],$ and optical matrix elements both give rise to the nonmonotonic behavior. Reported experimental measurements are also shown for comparison [10].

Figure 5

Intrinsic emittance of bulk PbTe(111) at 300 K calculated without any experimental scaling factors (blue), shifting the DFT calculated bands to match the experimental band gap of 0.19 eV [29] (orange), and fitting the bands to match the experimental band gap and electron effective mass of $0.24m_e$ (green) [31]. The reported experimental measurements of intrinsic emittance are also shown for comparison (red) [10].

Figure 6

Calculated variation in the intrinsic emittance of bulk PbTe(111) with respect to the work function $\varphi$. The precise value of the work function has considerable impact on the calculated emittance at the photoemission threshold, but is less significant at high photon energies. The experimentally measured work function of PbTe(111) is reported to be 4.0 eV [10] whereas the LDA calculated work function is reported to be 4.21 and 4.54 eV for Pb and Te terminated surfaces, respectively [18].

Figure 7

Effect of surface states on intrinsic emittance calculations. DFT calculated conduction-band states for a (a) Cu(100) surface slab and (b) bulk Cu. The conduction-band states which can contribute to the photoemission are highlighted in blue ($w_i\ne 0$ at $\hslash \omega \ge 5\mathrm{eV}$). (c) DFT calculated and experimental intrinsic emittance of Cu(100) at 300 K [32]. The experimentally observed low emittance at the photoemission threshold in Cu(100) must arise from surface states, due to the absence of bulk states directly above the vacuum level. (d) The experimentally measured intrinsic emittance for W(100) [33] is reasonably well described by both bulk and surface slab calculations, in contrast to Cu(100).

Figure 8

Intrinsic emittance for five candidate materials [Cu(100), Rh(110), PbTe(111), Mo(100), and W(100)]. We compare the analytical expression shown in Eq. (5), our DFT model, and previous experimental measurements [10, 32, 33, 40]. The Rh, Mo, and W measurements were obtained with a solenoid scan technique [40], whereas the Cu and PbTe measurements were obtained using a free-flight method [32]. We calculated the intrinsic emittance of Cu(100) and Rh(110) from their surface bands, whereas we calculated the intrinsic emittance of W(100), Mo(100), and PbTe(111) from their bulk bands. Experimentally, Rh(110) was observed to have a surface oxide layer during the intrinsic emittance measurements, whereas Cu(100) was measured with an atomically flat and clean surface. The ability of our model to also account for finite-temperature effects is also illustrated for Cu(100) at 300 and 30 K.

Figure 9

Mean absolute error (MAE) of the intrinsic emittance calculated with our DFT-based method and the analytical expression [Eq. (5)] over the photon energy range of the experimental measurements, shown in Fig. 8. The MAE is calculated with respect to experimental measurements [10, 32, 33, 40]. An approximate value of the experimental precision is also shown for Ref. [40].

Figure 10

Effect of $\sigma$ [Eq. (17)] on the calculated intrinsic emittance of bulk PbTe(111). We assume a lifetime broadening of $\sigma =0.20\mathrm{eV}$ for all intrinsic emittance calculations in this work.

Figure 11

Maximum excess energy ($E-V_0$) that prevents $G_{x,y}\ne 0$ terms of Eq. (29) from contributing to the photoemission of Rh(110) for (a) $G_x=\frac{2\pi }{a},G_y=0$ and (b) $G_x=0,G_y=\frac{2\pi }{b}$.

Figure 12

Minimum lattice constant (a) of general photocathode materials that allow for $G_x\ne 0$ to contribute to the photoemission, with $G_y=0,k_y=0$ [Eq. (31)].