First-principles study of structural, elastic, electronic, transport properties, and dielectric breakdown of Cs₂Te photocathode

Abstract

The pursuit to operate photocathodes at high accelerating gradients to increase brightness of electron beams is gaining interests within the accelerator community, particularly for applications such as free electron lasers (FEL) and compact accelerators. Cesium telluride (Cs₂Te) is a widely used photocathode material and it is presumed to offer resilience to higher gradients because of its wider band gap compared to other semiconductors. Despite its advantages, crucial material properties of Cs₂Te remain largely unknown both in theory and experiments. In this study, we employ first-principles calculations to provide detailed structural, elastic, electronic and transport properties of Cs₂Te. It is found that Cs₂Te has an intrinsic mobility of 20 cm²/Vs for electrons and 2.0 cm²/Vs for holes at room temperature. The low mobility is primarily limited by the strong polar optical phonon scattering. Cs₂Te also exhibits ultralow lattice thermal conductivity of 0.2 W/(m*K) at room temperature. Based on the energy gain/loss balance under external field and electron–phonon scattering, we predict that Cs₂Te has a dielectric breakdown field in the range from ~ 60 to ~ 132 MV/m at room temperature dependent on the doping level of Cs₂Te. Our results are crucial to advance the understanding of applicability of Cs₂Te photocathodes for high-gradient operation.

Introduction

In advanced photocathode community, there is a growing interest in applying high gradients to photocathodes to increase brightness of electron beams^1,2. One concern regarding the application of high gradients to semiconductor photocathodes is that they are susceptible to dielectric breakdown due to impact ionization^3,4. Impact ionization occurs when the electric field applied to a material exceeds a critical threshold, causing an abrupt increase in electrical conductivity. This phenomenon results from the acceleration of charge carriers in materials to velocities capable of creating additional free charge carriers through collisions. As a result, an avalanche process occurs, leading to a rapid increase in the number of free charge carriers and a significant decrease in the material’s resistance. Such breakdown can result in failure of the device or degradation of performance.

It is reasoned that semiconducting cesium telluride (Cs₂Te) photocathode may be capable of withstanding higher electric fields due to the larger band gap of 3.3 eV compared to other semiconducting materials, such as alkali antimonide. Cs₂Te is a popular photocathode material because of its high quantum efficiency (QE) exceeding 10% at its working wavelength, long lifetimes over a span of several months, stability under various operating conditions, and relatively fast response times⁵. Cs₂Te photocathode is in operational photoinjectors⁶ for several major accelerator facilities such as LCLS-II (SLAC, USA), FLASH (DESY), and European XFEL (Germany), as well as in many smaller accelerators and test facilities worldwide⁷. At Los Alamos National Laboratory (LANL), we selected to use Cs₂Te to generate ultra-bright electron beam at the Cathodes and Radiofrequency Interactions in Extremes (CARIE) photocathode test stand⁸.

Despite its widespread use, many material properties of Cs₂Te relevant to its interactions with strong electromagnetic fields remain unknown. For instance, to the best of our knowledge, the electrical and thermal conductivities of Cs₂Te have never been reported in the literature. The electrical conductivity is crucial for understanding photocathode behavior, as poor electrical conductivity may result in ionized surfaces, impacting photoemission or even integrity of the photocathode⁹. Thermal conductivity is essential for heat dissipation during operation of photocathodes. Furthermore, the performance of Cs₂Te photocathodes under high gradients remains unclear and requires detailed understanding of the transport properties. Additionally, one quantity that is absolutely necessary to study is the dielectric breakdown field that Cs₂Te photocathodes can withstand and its dependence on RF frequency of the electromagnetic field.

In this study, we employ first-principles calculations based on density functional theory (DFT) and density functional perturbation theory (DFPT) to predict fundamental properties of Cs₂Te, including structural, elastic, vibrational, electronic and transport properties. We compute the electron–phonon and phonon–phonon scattering rates of Cs₂Te for the first time using first-principles approach, enabling the calculations of electron and phonon transport via the Boltzmann transport equation (BTE). Our calculations reveal that Cs₂Te has an ultralow thermal conductivity of 0.2 W/(m*K) due to its low group velocity, indicating that heat dissipation in Cs₂Te photocathodes will be inefficient, potentially impacting high power applications. We predict room-temperature intrinsic mobilities of 20 cm²/Vs for electrons and 1.2 cm²/Vs for holes, primarily limited by the strong polar optical phonon scattering. Additionally, using the von Hippel low-energy criterion¹⁰ the electron–phonon scattering rates allow us to estimate that the breakdown field varies in between 60 and 132 MV/m at room temperature depending on the doping level of Cs₂Te. These calculated results provide critical inputs for modeling Cs₂Te photocathodes, particularly regarding detailed photoemissive properties based on Monte Carlo simulations^11,12,13, which are key to advance understanding of applicability of Cs₂Te photocathodes in high-gradient accelerators.

Crystal structure of Cs₂Te

To determine the crystal structure and atomic positions in Cs₂Te, we fully relaxed the lattice parameters and atomic coordinates using the Vienna ab-initio simulation package (VASP)^1,2 based on DFT. The Perdew-Burke-Ernzerhof (PBE)¹⁴ functional of the generalized gradient approximation (GGA) was used for the electron exchange–correlation. Standard PAW-PBE pseudopotentials¹⁵ were used with a cutoff energy of 500 eV for the plane waves. The structure was converged with the maximum force on each atom smaller than 0.01 eV/Å. Cs₂Te belongs to the Pnma space group (no. 62) with an orthorhombic unit cell characterized by the three lattice vectors a, b, and c. The relaxation yielded lattice constants of a = 9.62 Å, b = 5.89 Å, c = 11.63 Å. The comparison between the computed and measured lattice constants is shown in Table 1^16,17. It is seen that the predicted lattice constants using PBE function are in good agreement with experimental data, within 2%. In Cs₂Te, Cs atoms occupy two inequivalent sites with different coordination and bond character, denoted as Cs_I and Cs_II in Fig. 1a. Cs_I is bonded with 5 neighboring Te atoms with bond length smaller than 4.5 Å. Cs_II is bonded with 4 Te atoms with a bond length smaller than 3.9 Å. The average Cs-Te bond length in Cs₂Te is approximately 3.95 Å. Alkali metal tellurides have been reported in anti-CaF₂-type structures¹⁸; however, Cs₂Te in the anti-CaF₂-type structure is less stable and has higher energy compared to the Pnma structure¹⁹. Figure 1b shows the Brillouin zone and high-symmetric points used for phonon and band structure calculations.

Table 1 Calculated and experimental lattice constants of Cs₂Te.

Fig. 1

(a) Crystal structure of Cs₂Te where the larger atoms are Cs and the smaller atoms are Te. Cs_I and Cs_II denote Cs atoms that occupy two inequivalent sites. (b) The schematic of the Brillouin zone used in calculating the electronic band structure and the phonon dispersion. Rendered using VESTA²⁰.

Elastic properties of Cs₂Te

Elastic properties are crucial to know in numerous technological applications related to mechanical properties of materials. While a crystal theoretically has 36 elastic constants, the inherent symmetry of the crystal significantly reduces the number of independent elastic constants. Cs₂Te has eight independent elastic constants: C₁₁, C_12, C₁₃, C₂₂, C₃₃, C₄₄, C₅₅ and C₆₆. These elastic constants can be averaged by integration on the unit sphere to calculate the standard Young modulus, bulk modulus, shear modulus, and Poisson’s ratio. Utilizing the ELATE code²¹, we computed the elastic properties of Cs₂Te, which are presented in Table 2. The elastic constants clearly satisfy the Born stability criteria²² for materials with orthorhombic structures: (i) C₁₁ > 0; (ii) C₁₁*C₂₂ > C₁₂²; (iii) C₁₁*C₂₂*C₃₃ + 2C₁₂*C₁₃*C₂₃ − C₁₁*C₂₃² − C₂₂*C₁₃² − C₃₃*C₁₂² > 0; (iv) C₄₄ > 0; (v) C₅₅ > 0; (vi) C₆₆ > 0. The calculated bulk modulus and shear modulus are 9.52 GPa and 3.72 GPa, respectively. The small bulk and shear moduli indicate Cs₂Te is a relatively soft material, likely due to the large Cs-Te bond length that leads to weak chemical bonds as described in the previous section.

Table 2 Calculated elastic constants, bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio ν of Cs₂Te.

Vibrational properties of Cs₂Te

We investigated the vibrational properties of Cs₂Te using density functional perturbation theory (DFPT). To ensure the convergence of the vibrational frequencies, we constructed a 1 × 3 × 1 supercell for phonon dispersion calculations. A Monkhorst–Pack k-mesh of 2 × 1 × 2 was used to sample the Brillouin zone (BZ). Harmonic second-order interatomic force constants (IFCs) were derived using the finite displacement method, utilizing force constants obtained from VASP calculations. Subsequently, we calculated the phonon dispersion of Cs₂Te using the PHONOPY package based on these IFCs²⁶. Our calculated phonon dispersion, shown in Fig. 2, reveals absence of imaginary frequencies, indicating the dynamical stability of the structure. The highest vibrational frequency is observed to be approximately 100 cm⁻¹ at the Γ-point. Moreover, a discontinuity in the phonon dispersion near the Γ-point is evident, which is caused by the anisotropy of crystal structure. In our calculations, we considered the non-analytic term corrections, addressing the influence of the polarization field on the optical phonon behavior around the Γ-point. The strong anisotropy of the polarization along different directions leads to direction-dependent corrections, resulting in the discontinuity of the phonon dispersion at the Γ-point when the phonon wave vector changes direction.

Fig. 2

Phonon dispersion and phonon density of states of Cs₂Te.

The vibrational frequencies of each phonon mode at the Γ-point are listed in Table 3. The Raman and Infrared activity of each mode are also shown in the table. There are 18 Raman active mode (6Ag + 3B1g + 6B2g + 3B3g) and 12 Infrared active modes (5B1u + 2B2u + 5B3u). These computed vibrational frequencies can be compared with experimental Raman/Infrared spectra to validate the quality of the grown Cs₂Te films.

Table 3 Phonon frequencies of Cs₂Te at the center of Brillouin zone (Γ-point), and their Raman/Infrared activity.

Lattice thermal conductivity of Cs₂Te

Thermal conductivity plays a pivotal role in managing heat within high-power devices and is of course important to know to understand heat transport in photocathodes. To compute lattice thermal conductivity, we used the anharmonic third-order IFCs. Utilizing the same 1 × 3 × 1 supercell and 2 × 1 × 2 Monkhorst–Pack k-mesh as in Section IV, we got these anharmonic third-order IFCs, considering interactions up to fourth nearest neighbors. Additionally, we obtained the dielectric tensor and Born effective charges based on DFPT to account for long-range electrostatic interactions. The high-frequency dielectric tensor exhibits slight anisotropy with εxx = 4.09, εyy = 4.06, and εzz = 4.13. The computed diagonal Born effective charges are (1.21, 1.30, 1.15) for Cs_I, (1.04, 0.96, 1.12) for Cs_II, and (− 2.26, − 2.25, − 2.28) for Te in units of electron charge. The off-diagonal components are lower than 0.1 and not reported here. The positive Born effective charge for Cs atoms and the negative Born effective charge clearly indicate the ionic nature of the bonds in Cs₂Te. The intrinsic lattice thermal conductivity can be calculated by solving the phonon BTE as implemented in ShengBTE code²⁷ that considers the phonon–phonon scattering processes.

The calculated lattice thermal conductivity of Cs₂Te is shown in Fig. 3a, revealing slight anisotropy along different directions. As temperature increases, phonon scattering intensifies, leading to a decrease in lattice thermal conductivity. At 300 K, the average thermal conductivity of Cs₂Te is predicted to be ~ 0.2 W/(m*K), calculated as the average of thermal conductivities along the three crystallographic axes. This low lattice conductivity places Cs₂Te among the materials with ultralow lattice thermal conductivity²⁸. In comparison, Cu photocathodes have a thermal conductivity of ~ 401 W/(m*K) at room temperature²⁹. The low thermal conductivity of Cs₂Te photocathodes implies that heat generated in the photocathodes, whether by illuminating lasers or Joule heating, cannot be efficiently dissipated as in Cu photocathodes. This limitation may lead to localized heating, potentially affecting the quality of the generated electron beams. This should be studied experimentally, especially when the photocathode gun operates under conditions that generate high-brightness electron beams at high gradients.

Fig. 3

(a) Lattice thermal conductivity, (b) cumulative lattice thermal conductivity, (c) group velocity, and (d) phonon scattering rate of Cs₂Te at 300 K. The acoustic and optical modes are illustrated with blue and orange stars, respectively.

The cumulative thermal conductivity as a function of phonon vibrational frequency of Cs₂Te at 300 K is shown in Fig. 3b. It is seen that phonons with frequencies below 20 cm⁻¹, namely the acoustic phonons, contribute to nearly 50% of the lattice thermal conductivity. Around 20% of thermal conductivity is contributed by phonons with vibrational frequency in the range of 20–40 cm⁻¹, where acoustic and optical phonons interact. Therefore, the thermal conductivity of Cs₂Te is mainly determined by acoustic phonons and low frequency optical phonons. The maximum mean free path (MFP) of phonons is close to 100 nm and phonons with MFP less than 30 nm contribute 70% to the thermal conductivity. To further clarify the microscopic mechanism behind the ultralow lattice thermal conductivity for Cs₂Te, we plotted the group velocity and phonon–phonon scattering rates (Fig. 3c, d). The plots show that low-frequency acoustic phonons exhibit higher group velocity and lower scattering rates, while optical phonons have high scattering rates and low group velocities, thus contributing less to thermal conductivity. Therefore, the overall low lattice thermal conductivity results from the heavy atomic mass of Cs, and the relatively weak chemical bonds between Cs and Te as seen from the small bulk and shear moduli of Cs₂Te in Section III.

Electronic structure of Cs₂Te

The calculated band structure of Cs₂Te with GGA-PBE functional is shown in Fig. 4a. The band gap is predicted to be ~ 1.7 eV, notably lower than the experimental value of ~ 3.3 eV³⁰. Our results align with previous DFT calculations³¹. It is acknowledged that the GGA-PBE functional tends to underestimate band gaps of semiconducting materials³². However, despite this limitation, the band dispersion obtained with GGA-PBE functional is similar to that calculated from more advanced functionals, such the HSE06 hybrid functional³¹. Therefore, we anticipate that GGA-PBE functional accurately captures the effective masses and group velocities of electrons and holes, which determine their transport properties. From the band dispersion, the calculated effective masses for electrons are 0.267 ${m}_{0}$, 0.270 ${m}_{0}$, and 0.267 ${m}_{0}$ along the Γ–X, Γ–Y, and Γ–Z directions, respectively, where ${m}_{0}$ is the rest mass of an electron. Much larger effective masses are observed for holes, which are − 2.524 ${m}_{0}$, − 0.732 ${m}_{0}$, − 1.900 ${m}_{0}$, along the Γ–X, Γ–Y, and Γ–Z directions, respectively.

Fig. 4

(a) Electronic band structure, and (b) Burstein-Moss energy shift as a function of carrier concentration of Cs₂Te. Positive shifts for electrons, and negative shifts for holes.

The change of the Fermi level as a function of carrier concentration due to Burstein-Moss effect can be calculated using^33,34

$$\Delta E=\frac{{h}^{2}}{8{\pi }^{2}{m}^{*}}{(3{\pi }^{2}n)}^{2/3},$$

(1)

where $h$ is Planck’s constant, and $n$ is the concentration of charge carriers (electrons or holes) associated with the doping level of semiconductors. The energy shifts caused by excess electrons are calculated relative to the conduction band minimum (CBM), and those induced by excess holes are referenced to the valance band maximum (VBM). Note that the shift of Fermi level can be caused by doping the materials with either intrinsic defects or extrinsic dopants. Figure 4b depicts the shifts of the Fermi level as a function of carrier concentration of electrons (n-type) and holes (p-type) in Cs₂Te. With an electron concentration of 10²¹ cm⁻³, the Fermi level is observed to shift 0.8 eV above CBM, whereas with a same hole concentration, it shifts only 0.4 eV below VBM. This difference arises from the narrower dispersion of the valance bands compared to the conduction bands, as evidenced from the band structure. The transport properties of electrons and holes are evaluated within the carrier concentration range of 10¹⁸ cm⁻³ to 10²¹ cm⁻³.

Electron–phonon scattering and carrier mobility in Cs₂Te

To gain fundamental understanding of the electron transport properties in Cs₂Te at high electromagnetic fields, we investigated the electron–phonon scattering. The electron transport properties were calculated with the ab-initio scattering and the transport method implemented in the AMSET code^35,36. The package solves the electronic BTE in the momentum relaxation-time approximation (MRTA) to calculate scattering rates and mobilities within the Born approximation. In AMSET code, the relaxation time is estimated by the number of different band and k-point dependent scattering rates instead of the constant relaxation-time approach. In this work, we calculated scattering rates by considering the following scattering processes: (a) acoustic deformation potential (ADP), which is responsible for the phonon-electron interactions; (b) ionized impurity (IMP) scattering, which represents scattering of charge carriers by ionization of the lattice; and (c) polar optical phonon (POP) scattering, which considers the interaction between polar optical phonons and electrons. The resulting carrier relaxation time, namely the inverse of the scattering rate $\tau$, was calculated by Matthiessen’s rule³⁵

$$\frac{1}{\tau }=\frac{1}{{\tau }^{ADP}}+\frac{1}{{\tau }^{IMP}}+\frac{1}{{\tau }^{POP}},$$

(2)

where ${\tau }^{ADP}$, ${\tau }^{IMP}$, and ${\tau }^{POP}$ are the relaxation times from the ADP, IMP, and POP scattering, respectively. In ADP and IMP scatterings, electrons do not acquire or lose energy while undergoing elastic scattering, whereas POP scatterings exhibit inelastic scattering that occurs due to phonon emission or absorption. The Fermi’s golden rule is used to calculate the scattering rates for elastic and inelastic scattering from an initial |nk > state to the final |mk + q > state. The scattering rate can be written as.

$${\tau }_{n{\varvec{k}}\to m{\varvec{k}}+{\varvec{q}}}^{-1}=\frac{2\pi }{\hslash }{\left|{g}_{nm}\right|}^{2}\delta \left({\epsilon }_{n{\varvec{k}}}-{\epsilon }_{m{\varvec{k}}+{\varvec{q}}}\right),$$

(3)

where ${\epsilon }_{n{\varvec{k}}}$ and ${\epsilon }_{m{\varvec{k}}+{\varvec{q}}}$ are the energy of the initial state and the final state after scattering, ${g}_{nm}$ is the electron–phonon coupling matrix element that is defined as.

$${g}_{nm}\left({\varvec{k}},{\varvec{q}}\right)=\langle m{\varvec{k}}+{\varvec{q}}|{\Delta }_{q}V|n{\varvec{k}}\rangle ,$$

(4)

where ${\Delta }_{q}V$ is the electronic perturbation from different scattering mechanisms. The required parameters to obtain the scattering rates, such as dense and uniform band structures, elastic coefficients, wave-function coefficients, deformation potentials, static and high-frequency dielectric constants, and polar-phonon frequency, were calculated from DFT and DFPT.

The calculated electron and hole scattering rates at 300 K are shown in Fig. 5a, b. The figures show that the POP scattering clearly dominates due to the pronounced polarity of the chemical bonds in Cs₂Te. The ADP scattering and IMP scattering rates have much smaller contributions to the total scattering rates. The electron scattering rate is around 450 ps⁻¹ near the CBM (Fig. 5b). The hole scattering rate at VBM is around 850 ps⁻¹. These calculated scattering rates were used to predict the carrier transport and the breakdown field of Cs₂Te.

Fig. 5

Electron–phonon scattering rates for (a) p-type carrier (holes), and (b) n-type carrier (electrons) of Cs₂Te at 300 K. The carrier concentration is 10¹⁸ cm⁻³. Black dots are the total scattering rates. Green, orange, and blue dots represent the contribution from POP, ADP, and IMP scattering, respectively. The Fermi level is set to 0 eV. Negative energy in (a) is related to holes, and positive energy in (b) is related to electrons. A scissor correction is added to shift the CBM to experimental value of 3.3 eV.

The calculated hole and electron mobilities as functions of temperature and carrier concentrations are shown in Fig. 6. In Cs₂Te with an electron concentration smaller than 10¹⁸ cm⁻³, the hole mobility is ~ 2.0 cm²/(V*s) at 300 K (Fig. 6a), and the electron mobility is around 20 cm²/(V*s) (Fig. 6b). Both electron and hole mobilities slightly decrease with carrier concentrations up to 10²⁰ cm⁻³, and rapidly decrease when the carrier concentration is larger than 10²⁰ cm⁻³. Hole and electron mobilities also decrease with temperature as seen from Fig. 6c, d, due to the increased scattering rates at higher temperatures. The much lower mobility of holes than that of electrons is due to the larger scattering rates and the large effective masses of holes in Cs₂Te.

Fig. 6

(a, b) Carrier mobility as a function of carrier concentration for p-type (holes) and n-type (electrons) carriers in Cs₂Te at 300 K. (c, d) Carrier mobility as a function of temperature with a carrier concentration of 10¹⁸ cm⁻³.

Figure 7a, b show calculated electrical conductivities of p-type and n-type doped Cs₂Te. Both p-type and n-type conductivities increase monotonously with carrier concentration from 10¹⁸ cm⁻³ to 10²¹ cm⁻³. The increase of conductivities with carrier concentrations is due to the increase in the number of carriers participating in the conduction process. The electrical conductivities decrease with temperature, which can be attributed to the increased electron–phonon scattering rates at higher temperatures. Moreover, it is seen that the n-type conductivity is around one order of magnitude larger than the p-type conductivity at the same carrier concentration and temperature.

Fig. 7

Electrical conductivity of (a) p-type and (b) n-type Cs₂Te as a function of carrier concentrations at different temperatures.

Dielectric breakdown in Cs₂Te

The dielectric breakdown field of Cs₂Te is estimated via a fully first-principles approach based on the Frohlich-von Hippel criterion^10,37,38. This method contemplates that an electron avalanche occurs when the electron energy reaches the threshold to excite a valence electron across the electronic energy gap, causing electron multiplication and leading to impact ionization. Dielectric breakdown occurs when the rate of energy gain $A\left(E,\epsilon ,T\right)$ of an electron with energy $\epsilon$ due to the external field $E$ at temperature $T$ is larger than the energy-loss rate $B\left(\epsilon ,T\right)$ due to electron–phonon scattering.

$$A\left(E,\epsilon ,T\right)>B\left(\epsilon ,T\right),$$

(5)

for energies $\epsilon$ ranging from CBM to the impact ionization threshold of CBM + E_g, where E_g is the band gap of the material^10,38,39,40. The energy-gain rate of the electron can be evaluated as.

$$A\left(E,\epsilon ,T\right)=\frac{1}{3}\frac{{e}^{2}\tau (\epsilon ,T)}{{m}^{*}}{E}^{2},$$

(6)

where $e$ and ${m}^{*}$ are the electronic charge and effective mass, respectively. $\tau (\epsilon ,T)$ is the energy- and temperature-dependent electron–phonon scattering rate, as shown in Fig. 5b. The field-independent rate of energy loss $B\left(\epsilon ,T\right)$ to the lattice is obtained by subtracting the rate of phonon absorption from phonon emission³⁸

$$B\left(\epsilon ,T\right)=\frac{2\pi }{\hslash D\left(\epsilon \right)}\sum_{\pm }\sum_{n{\varvec{k}}}\sum_{m{\varvec{k}}+{\varvec{q}}}\left\{\begin{array}{c}\pm {\hslash \omega }_{q}{\left|{g}_{nm}\left({\varvec{k}},{\varvec{q}}\right)\right|}^{2}\left({n}_{{\varvec{q}}}+\frac{1}{2}\mp \frac{1}{2}\right)\\ \times \delta \left({\epsilon }_{n{\varvec{k}}}-{\epsilon }_{m{\varvec{k}}+{\varvec{q}}}\pm \hslash {\omega }_{{\varvec{q}}}\right)\delta \left({\epsilon }_{n{\varvec{k}}}-\epsilon \right)\end{array}\right\},$$

(7)

where ${n}_{{\varvec{q}}}$ is the phonon occupation number that is given by Bose–Einstein distribution, $\pm$ sign represents the adsorption and emission of phonon, respectively. ${\omega }_{{\varvec{q}}}$ is the phonon frequency. We used an effective phonon frequency that is defined as the weighted sum over all phonon modes at Γ-point³⁵. In Fig. 8a we present the energy loss rate as a function of energy (black line) and energy gain rate under different external fields (blue lines), starting from the CBM. The intrinsic breakdown field is defined as the smallest external electric field such that the energy gain rate is larger than the energy loss rate for all energies between the CBM and the CBM plus the band gap (3.3 eV). We found that that the breakdown field is ~ 60 MV/m at room temperature for Cs₂Te with an electron concentration of 10¹⁸ cm⁻³. Using the same approach, we obtained the breakdown fields for different carrier concentrations and temperatures as shown in Fig. 8b. The breakdown field clearly increases with temperature and carrier concentrations due to the increased energy loss rate at higher temperatures and higher carrier concentrations. The dielectric breakdown field of Cs₂Te at 300 K is predicted to vary from 60 MV/m to 132 MV/m depending on the doping level of samples. It is to be noted that the von Hippel low-energy criterion should be seen as a lower bound for the breakdown field. This is because Cs₂Te is not available in single-crystalline form since it is grown via deposition methods that give rise to polycrystalline samples with varying grain sizes. The scattering introduced by the grain boundaries may further contribute to energy loss rate and increase the breakdown field. Additionally, the method considers the breakdown due to an applied DC field, which may not be suitable in conditions in RF photoinjectors where photocathodes are exposed to RF fields.

Fig. 8

Dielectric breakdown of Cs₂Te. (a) Energy-gain rate $A\left(E,\epsilon ,T\right)$ from an applied external field and energy loss rate $B\left(\epsilon ,T\right)$ due to electron–phonon scattering. Both quantities are calculated at 300 K for an electron concentration of 10¹⁸ cm⁻³. The intrinsic breakdown occurs when the applied electric field is such that the energy gain rate is larger than the loss rate for all energies between the conduction-band minimum (CBM) and the CBM plus the energy of the band gap (3.3 eV). (b) The dielectric breakdown field of Cs₂Te as a function of temperature for doping concentration of 10¹⁸ cm⁻³ and 10²¹ cm⁻³.

Conclusion

This paper reports results of our investigation of the structural, elastic, electronic, vibrational, and transport properties of Cs₂Te using first-principles calculations. We computed the electron–phonon and phonon–phonon scattering rates of Cs₂Te using first-principles approach for the first time, enabling calculations of electron and phonon transport via the Boltzmann transport equation (BTE). We computed material properties of Cs₂Te, such as lattice thermal conductivity, carrier mobilities and conductivities, based on the electron–phonon and phonon–phonon scattering. We predicted the intrinsic electron and hole mobility of Cs₂Te to be 20 and 1.2 cm²/Vs, respectively, at room temperature. The low mobility is explained by the strong polar optical phonon scattering. Cs₂Te is also predicted to have ultralow lattice thermal conductivity of 0.2 W/(m*K) at room temperature due to low group velocities and relatively weak bonds in Cs₂Te. Based on the electron–phonon scattering rates and the von Hippel low-energy criterion, the dielectric breakdown field of Cs₂Te is predicted to vary between 60 MV/m 132 MV/m at room temperature, depending on the doping of the material. Our studies serve as a basis for understanding the high-gradient performance of Cs₂Te photocathodes.