Heating of $C s 2 Te$ photocathode via field emission and radiofrequency pulsed heating: Implication toward breakdown Open Access

Semiconductors, such as $C s 2 Te$⁠, are gaining interest as electron emitters in radiofrequency (RF) photoinjectors for high-power accelerators and scientific instruments, such as time-resolved electron microscopes and free electron lasers. Compared to metal cathodes, $C s 2 Te$ offers advantages, such as low intrinsic emittance and high quantum efficiency.^1,2 In photoinjectors, higher operating fields are desired to enhance beam brightness, which can lead to new research frontiers by enhancing the signal to noise ratio or temporal resolution. However, RF breakdown acts as a primary factor limiting the maximal operating fields that can be sustained. Breakdowns can significantly affect the structure and morphology of inner walls that face high electric fields (including a photocathode surface), eventually leading to either gradual or instant performance degradation.³ 

Experimental observations indicate that breakdowns are typically accompanied by dark current and temperature spikes. A hypothesis then can be formed that an elevated/spiking temperature is a direct consequence of the dark current spike through explosive field emission at nanoasperities.⁴ This triggers a feedback loop leading to a catastrophic thermal runaway on the surface, which can culminate in a fully developed breakdown event, ending with the formation of a vacuum arc and power quenching. Additionally, the frequency at which breakdowns occur is very sensitive to the RF magnetic field component mediated via so-called pulsed heating effects.⁵ While the underlying breakdown nucleation mechanisms are extremely complex, considerable progress has been made in unveiling various aspects of the breakdown initiation process in metals over the past decade;^6–12 to date, corresponding investigations in semiconductors have been comparatively lacking.

Recently, high-gradient C-band (⁠ $f r f∼5.7$ GHz) accelerators were identified as promising systems for industry, medicine, national security, and basic sciences. These systems strike a balance between being relatively compact (as compared to the L- or S-band) while still being able to transport high charge/high current particle beams with minimal wakefield/breakup instabilities (as compared to the X-band). Future technological requirements point to the following operating ranges: electric gradient $E$ up to 300 MV/m, magnetic field $H$ up to 500 kA/m, pulse length $t p$ between 0.4 and 1  $μ$s, and repetition rate $f r$ up to 200 Hz.^2,13 This pushes the material accelerator cavities that are made of into unprecedented conditions, requiring careful assessment in terms of breakdown susceptibility.

In this work, we investigate the relative contributions of both field emission and pulsed heating due to a high magnetic field on $C s 2 Te$-on-Cu photocathode heating to delineate the regimes that could potentially lead to a runaway breakdown process. Our findings suggest that field/dark emission has an overall limited contribution to the heating of the $C s 2 Te$ surface due to the charge depletion characteristic of any intrinsic or low doped semiconductor. In contrast, photocathode heating caused by the surface magnetic field was much stronger in the range of the studied electric field/magnetic field/repetition rate parameter. Hence, the presented results provide an avenue to refine and validate the commonly assumed mechanisms of the physics of breakdown.

A system comprised of an ultrathin 50 nm $C s 2 Te$ film grown on a Cu semi-infinite substrate within the described operating envelope was analytically modeled with respect to two key mechanisms: field-emission-induced heating and RF pulsed heating.

Although heat diffusion in the present work could be solved as a 1D problem, a 2D model was chosen to facilitate future model extensions, where coupling of the heating effects on surface evolution¹² becomes possible.

Experiments show that the field-emission current characteristic of a semiconductor significantly deviates from the classical Fowler–Nordheim (FN) theory,^14–17 which is commonly used to predict field emission from metals. Another theory that was developed well beyond the classical FN theory, known as Stratton–Baskin–Lvov–Fursey (SBLF) formalism,^18,19 proved to be indispensable when modeling field emission from semiconductors. Unlike in metals, when a semiconductor is placed in a strong enough electric field, that field penetrates to certain depth and depletes that region from charge. The charge can be moved across the system with a latency associated with the transit time across the depleted depth. It must be noted that SBLF theory does not include surface states analysis. In Ref. 20, using an example of H-passivated Si wires, the importance of competing effects between bulk depletion and surface state conductivities on current saturation was highlighted. Since little is known about Cs $2$Te surface defects/states, this issue remains beyond the scope of this work. Instead, this work attempts to first create a baseline result using the experimentally known basic band structure²¹ and density functional theory (DFT) computed electronic and transport properties.²² 

As shown in Fig. 2, in a 1D energy-coordinate representation, field penetration may cause the conduction band minimum (CBM) to bend below the Fermi level, allowing for an inversion layer to form and creating a well where electrons can accumulate near the surface of the semiconductor, thereby effectively “metallizing” the surface. The tunneling probability from that well then defines the initial phase of field-emission current, often appearing as if the current-field characteristic follows the conventional FN law. However, as the tunneling probability increases with the increase of the surface field, the electron well quickly depletes and new electrons have to be resupplied to sustain a high field-emission current. Current saturation is expected when field emission becomes resupply-limited due to (i) a limited amount of charge, (ii) a limited drift velocity, and (iii) distance/time to drift.

To describe this picture self-consistently, series equations must be solved together with special boundary conditions. A schematic energy diagram of the band bending region specific for a $p$-type semiconductor (such as $C s 2 Te$⁠) is shown in Fig. 2. Here, $E C$ and $E V$ are the CBM and the valence band maximum (VBM), respectively. $E F$ is the Fermi level, and $E A$ is the acceptor level. $E g$ is the bandgap, $E a$ is the energy of the acceptor level with respect to the CBM, $χ$ is the electron affinity, $φ$ is the work function, and $θ(x)$ defines the energy of the CBM with respect to the Fermi level.

Solving Eq. (3b) yields the relationship between the electric field and $y(x)$ for a given constant current $j$ flowing through the material. Since the semiconductor layer is ultrathin, it is reasonable to assume that the electric field penetrates the entire depth of the semiconductor and terminates at the Cu surface. The boundary condition $y b( x b)$ can then be determined from the value of the field $F b$ at the conductor-semiconductor boundary where $F b$ can be found by solving Ohm’s equation in the boundary as $F b=j/ [ q ( n ( y b ) μ e + p ( y b ) μ p ) ] − 1$⁠, which defines the relationship between the field and carrier transport at the interface.

The differential equation (3b) is solved throughout the entire semiconductor layer. However, due to the coordinate transformation into the dimensionless domain $y$⁠, the Poisson equation alone does not directly determine the electric field or band bending at the semiconductor-vacuum interface. Therefore, to obtain the field-emission current, one needs to simultaneously solve the Poisson equation and the Stratton equation (which gives the emission current density from the conduction band).¹⁹ This coupled solution self-consistently determines the surface band bending by resolving the relation between the electric field and the charge transport at the surface-vacuum interface.

When the emission current $j e m$ is held fixed, we can retrieve the band bending on the surface $y s$ as a function of the surface field $F s$⁠. With negative surface levels, $y( F s)<0$⁠, the Stratton equation can be written as a simplified Nordheim equation,^18,19,23

The $j−E$ curve is obtained from solving multiple sets of coupled Poisson and Stratton equations, where each set has a different value for the fixed $j$ and $j e m$⁠. The two equations give the electric field as a function of dimensionless variable $y$ where the intersection of the curves provide the current-density dependencies on the electric field as plotted in Fig. 3. Each intersection becomes a data point along the $j−E$ curve.

Notably, the discontinuity of Stratton’s equation at $y=0$ does not influence the calculated current, as the intersection inherently accounts for a charge redistribution at the semiconductor-vacuum interface. Moreover, the SBLF current profile ensures that the energy diagram evolves self-consistently with the changing RF electric field. Our approach dynamically solves Poisson and Stratton’s equations such that the band bending at the semiconductor-vacuum interface follows the oscillating RF field. The SBLF current profile was then mapped to the temporal electric field variation during a pulse as defined in Eq. (1).

The current profile retrieved from the SBLF theory is finally used to calculate:

(1) the Joule heating generated in the $C s 2 Te$ layer (⁠ $Ω S$⁠), where the volumetric heat source in Eq. (2) is $f(x,t)= ρ e j e m 2(E(t))$ with $ρ e$ being the electrical resistivity and (2) the Nottingham heating generated in the $C s 2 Te$ layer (⁠ $Ω S$⁠), where the volumetric heat source is $f(x,t)=Δ U e j e m(E(t))$ with $Δ U e$ being the electron energy change (release or absorption) at the vacuum- $C s 2 Te$ interface (see below).²⁴ 

Pulsed heating arises from the ohmic losses of surface eddy currents induced by the time-dependent RF magnetic field. Here, the pulsed heating is considered to act solely in the Cu subdomain because thin-film $C s 2 Te$ will be invisible to the tangential magnetic field. Indeed, in the GHz range, the magnitude of power dissipation in an insulator or a semiconductor is measured through the loss tangent calculated as²⁵  $tan⁡( δ d)= ε ″ ε ′≈ σ ω ε 0 ε r$ with $ε ″$ and $ε ′$ the material complex permittivity $ε c=ε−i σ ω= ε ′−i ε ″$⁠, and $δ d$ is the loss angle. A nonintentionally doped wideband gap semiconductor, such as $C s 2 Te$⁠, should naturally have high resistivity (⁠ $∼100Ω$m) and high relative permittivity (10.17),²⁶ leading to a very low estimate of the loss tangent on the order of $10 − 2$ under operating frequencies in GHz. From this, power dissipation and in turn heat generation in the semiconductor layer should be negligibly small. Indeed, once the loss tangent is taken into account, a rudimentary calculation with $ΔT= P a v e⋅ R t h e r m a l$ shows the relative temperature rise in the Cs $2$Te layer to be on the order of 1 K. Direct dissipation in the semiconductor was, thus, neglected in the following.

The heat diffusion equation was solved on a 2D domain using the finite-element method. This required turning Eq. (2) into its variational formulation and discretizing it under a time step $Δt$⁠. Let $t n$ denote the time at the nth time step with $T n$ being the temperature field at time $t n$⁠. Using the backward Euler method, the time derivative can be approximated as $∂ T ∂ t | t = t n= T n + 1 − T n Δ t$⁠. This equation can be written in the weak formulation as $a(T,v)= L n + 1(v)$ by multiplying the PDE with a test function $v∈ V ^(Ω)$ and integrating the second order derivative by parts in domain $Ω$⁠. If we consider both the Neumann boundary condition on $∂ Ω S V$ and the Robin boundary condition on $∂ Ω C$⁠, the following equation can be obtained:

This resulting governing equation was solved utilizing the open source PDE solver FEniCS.²⁸ FEniCS includes a domain-specific language called the Unified Form Language where the variational forms are defined. The governing equations were automatically compiled and executed in underlying C++ based computational kernel DOLFIN.²⁹ 

As explained in Sec. II C, the volumetric heat source on the Cu subdomain $Ω C$ is derived from Eq. (7). To simulate the heat transfer from the copper to the surrounding, a Robin boundary condition is implemented on the bottom of the copper domain to simulate convective cooling as

The boundary condition on the $Ω C/ Ω S$ interface was set in the weak formulation due to the integration over the entire domain, which naturally includes the shared interface, even in the presence of material discontinuities. For a system with two subdomains, $Ω C, Ω S$⁠, each with different material properties, the interface conditions are (i) temperature continuity (⁠ $T C= T s$⁠) and (ii) heat flux continuity (⁠ $κ C ∂ T ∂ n C= κ S ∂ u ∂ n S$⁠). Such continuity conditions are automatically satisfied within the finite element framework without requiring explicit enforcement.

The dependence of the current density on the surface field, calculated using the FN and SBLF formalism, is shown in Fig. 4(a). The results highlight a striking difference. A rapid current-density saturation at fields above 200 MV/m was observed within SBLF formalism. In contrast, predictions from the FN formalism (for a work function of 3 eV) predicted current densities 6–8 orders of magnitude larger for the same range of the surface field. It is also noted that there is a narrow range of fields (⁠ $j= 10 4$– $10 6$ A/m $2$⁠) where the FN and SBLF predictions are similar. This is the manifestation of the metal-like inversion layer discussed above. While the field-emission current profile of $C s 2 Te$ is currently not known in detail, we can look at other similar materials to validate our results. Chubenko et al. utilized a modified SBLF formalism to study field emission from $n$-type ultrananocrystalline diamond films, which showed good agreement with experiments.¹⁸ Field emission from $p$-type Si tip emitters was well characterized, e.g., by Serbun et al.¹⁷ Additional $j−E$ curves in Fig. 4(a) are theoretical SBLF calculations for Si emitters as described in Ref. 17. Here, the saturation current density calculated by the SBLF theory is a much better predictor than the conventional FN theory. With a rough estimate of dividing the current by the total surface area of the Si-tips, we arrive at an approximate experimental current-density saturation of $∼1× 10 7$ A/m $2$ where classical FN theory will massively overestimate the current density by multiple orders of magnitude. Similar current-density saturation levels are observed with atomically sharp silicon tips, where the current density is approximately $5.5× 10 8$ A/m $2$⁠,³³ which is much lower than the value associated with the vacuum space-charge saturation.

Figure 4(b) also demonstrates that the current saturation level is strongly dependent on the mobility of charge carriers, with higher mobilities resulting in higher saturation currents. This is a natural result of the SBLF-based model as saturation is linked to the efficiency of a charge drift through the depleted region resulting in the relatively low current density of $C s 2 Te$⁠.

The individual transient contributions of Joule and Nottingham effects to $C s 2 Te$ heating are displayed in Fig. 5. The results show that the Nottingham effect dissipates about an order of magnitude more power than Joule heating. This is expected under low (due to charge depletion saturation) current density conditions,³⁴ as Nottingham power scales as $j$ and Joule power scales as $j 2$⁠.

The combined contribution of pulse heating and field emission (including both Joule and Nottingham contributions) on the temperature evolution in the material can now be investigated. The input parameters used for these simulations are summarized in Tables I and II.

Figure 6(a) reports the temperature evolution due to a single RF pulse on the Cu/ $C s 2 Te$ boundary (black) and the $C s 2 Te$/vacuum boundary (blue). For a surface electric field of 300 MV/m and a magnetic field of 400 kA/m, the photocathode system reached a maximum $ΔT$ of $∼$40 K at the $C s 2 Te$/vacuum boundary and of $∼$30 K at the Cu/ $C s 2 Te$ boundary. After the end of the pulse, the system rapidly (⁠ $∼10μ$s) cools down to the initial temperature of the surrounding.

In a striking comparison, Fig. 6(b) reports the temperature rise when field emission is instead described using FN theory. In this case, the system would undergo a catastrophic failure due to an explosive temperature rise. This is consistent with the scenario described by Kyritsakis et al.³⁵ for sharp Cu emitters. Capturing the current saturation behavior accurately is, therefore, critical to correctly estimate the heat dissipation in this system and, hence, to realistically predict its stability under high RF power conditions.

Figure 6(c) reports the temperature distribution throughout the entire computational domain at $t=400$ ns when the pulse is turned off. The $C s 2 Te$/vacuum boundary is located at $x=60μ$m, while the Cu/ $C s 2 Te$ boundary is located at $x=59.95μ$m. The differences in the volumetric heat capacity and the thermal conductivity between the two materials results in a kink in the temperature profile at the metal/semiconductor interface. The results show that a temperature increase is particularly significant in the topmost $∼10μ$m closest to the vacuum interface.

To gain a better understanding of the individual contributions to the overall heating, a set of simulations was conducted where the applied magnetic field was varied from 0 to 500 kA/m while keeping the electric field at 0 V/m, and another set of simulations where the applied electric field was varied from 0 to 2000 MV/m while keeping the magnetic field at 0 A/m. The results are presented in Fig. 6(d). Both curves exhibit significantly different scaling behaviors. The saturation of the electric-field driven field-emission currents predicted by the SBLF theory leads to peak temperature increases between 10 and 30 K between about 225 MV/m and 2 GV/m and to very low heating below. In contrast, pulse heating driven by magnetic fields results in a quadratic increase in heat dissipation, leading to a crossover between electric/field emission-dominated regimes at low magnetic fields and a magnetic/pulse-heating-dominated regime at high magnetic fields.

Figure 7 shows the resulting temperature evolution at the $C s 2 Te$/vacuum boundary utilizing this time-averaged strategy, shown with a blue dashed line, which can be compared to a shorter explicit calculation without intrapulse averaging shown in orange, showing very good agreement. Extending the calculation until a steady state is reached leads to a predicted temperature increase of around 60 K.

The simulations presented above suggested that RF pulsed heating combined with field-emission heating would lead to relatively limited temperature elevations on the semiconductor photocathode surface. Additionally, due to field-emission current saturation, processes leading to runaway and catastrophic failure appear significantly less efficient in semiconductors than in metals where saturation of the field-emission currents is not expected. However, further analysis may be required to understand the emission characteristic for high-gradient application of semiconductor photocathodes where operational fields exceed 100 MV/m.

Field emission from semiconductors is indeed known to experience current saturation when the surface field is increased beyond a certain point, which limits heating contributions from field emission. However, the appearance of a different regime at ultrahigh gradients should also be considered. In DC experiments, it was indeed shown that at progressively higher surface fields exceeding the saturation point, the $I−V$ can switch from saturation to a highly nonlinear regime¹⁷ where the field-emission current increases rapidly. It was hypothesized that this rapid increase in electron emission results from thermionic emission,³⁷ pointing toward the possibility of a prethermal runaway/explosive emission stage. Another alternative explanation is that dielectric breakdown could lead to a similar increase in field emission when the local fields in the material become large enough to trigger avalanches of impact ionization, leading to the formation of a hot free carrier gas and to the restoration of FN-like field-emission behavior.

This study computationally investigated the thermal balance in a thin $C s 2 Te$ photocathode film on a metal operating under very high fields in order to assess its susceptibility to conventional thermal runaway leading to an electrical breakdown in vacuum. It was found that under planned upgraded application conditions, beyond the state of the art 100–150 MV/m, thermal runaway is impeded by field-emission current saturation. At the same time, it was found that the averaged pulse heating is also unlikely to yield catastrophic material failure.

However, peak temperature spikes and steady-state temperature rise can be expected between 70 and 100 K. Such temperature elevation on the photocathode surface, when taken together with very high electric fields, was recently shown to generate a non-negligible thermoelastic driving force that can result in diffusive roughening of the surface.¹² Previous research indicates that such a thermal-mechanical effect assists in the nucleation of the geometrical RF-breakdown precursors on metal surfaces and could potentially also be of importance to metal-coated semiconductor electron emission cathodes. Modeling of metal-semiconductor thermoelastic in high gradients is underway.

Additionally, the present work highlights the risk of avalanche breakdown of $C s 2 Te$ that could potentially lead to metal-like thermal runaway,⁶ subsequently triggering RF breakdown. Time-resolved multiphysics modeling of dielectric breakdown,³⁹ which could potentially become a limiting factor in a very high-gradient semiconductor cathode system, is being developed.

Ryo Shinohara, Soumendu Bagchi, Evgenya Simakov, and Danny Perez were supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under Project No. 20230011DR. The work by Sergey Baryshev was supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Award No. DE-SC0020429. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy (Contract No. 89233218CNA000001).

The authors have no conflicts to disclose.

Ryo Shinohara: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Soumendu Bagchi: Conceptualization (equal). Evgenya Simakov: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Writing – review & editing (equal). Danny Perez: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Sergey V. Baryshev: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.