Electron Trajectories and Critical Current in a Two-Dimensional Planar Magnetically Insulated Crossed-Field Gap

The critical current in a one-dimensional (1D) crossed-field gap is defined by the transition from a cycloidal flow to a near-Brillouin (nB) state characterized by electron flow orthogonal to both the electric and magnetic fields and uniform virtual cathode formation. Motivated by recent studies on space-charge-limited current in non-magnetic diodes, we assess the meaning of critical current in a magentically insulated two-dimensional (2D) planar crossed-field geometry. Particle-in-cell (PIC) simulations demonstrate that binary behavior between a laminar and turbulent state does not occur in 2D because the virtual cathode is nonuniform. Rather than a distinct nB state above the critical current as in 1D, there is an increase in Brillouin contribution with the presence of cycloidal components and noise even at low currents. To evaluate the electron flows in a 2D crossed-field gap in the absence of a binary transition, we developed two metrics to assess the Brillouin and cycloidal components in a 2D planar crossed-field gap for various emission widths and injection current densities by comparing the phase space plots from PIC simulations to analytical solutions for cycloidal and Brillouin flow. For a smaller emission width, less Brillouin contribution occurs for a given injection current, while maximizing the cycloidal noise requires a larger injection current. Once the virtual cathode starts to form and expand with increasing injection current, the cycloidal noise reaches its peak and then decreases while the Brillouin components become significant and increase.


I. INTRODUCTION
Crossed-field devices (CFDs) with orthogonal electric and magnetic fields, such as magnetrons and crossed-field The associate editor coordinating the review of this manuscript and approving it for publication was Wenxin Liu .amplifiers (CFAs), have been widely used in many applications, including high-power microwave generation, radar systems, communication, plasma generation for semiconductor processing, and nuclear fusion [1], [2], [3], [4], [5], [6], [7], [8], [9].CFDs often operate under magnetic insulation, meaning that an electron emitted from the cathode turns back to the cathode before reaching the anode [6].Magnetic insulation requires that the transverse magnetic field B exceeds the threshold of the Hull cutoff magnetic field [10], given by where m e and e are the electron mass and charge, respectively, V g is the gap voltage, v 0 is the electron emission velocity, and D is the gap distance.This threshold corresponds to the magnetic field at which an electron emitted in the x-direction (parallel to the electric field and perpendicular to the magnetic field) reaches the anode with zero-velocity in the x-direction.
Another important condition for crossed-field devices is the Buneman-Hartree (BH) condition, which requires the velocity of the electrons at the top of the electron Brillouin flow hub to match the phase velocity of the radio frequency (RF) field [11].The cavities on the anode (''slow-wave structure'' [4]) may be used to reduce the phase velocity of the RF field to ensure the BH condition or continuous synchronous interaction between the electrons and RF field for an oscillating state [6], [11].Theoretical and simulation studies of planar CFDs are more mature than cylindrical CFDs due to their relative simplicity, despite the more common use of the latter geometry in practical applications [7].Here, we focus on direct current (DC) CFDs in a planar geometry without cavities (e.g., with a smooth-bore anode [6]).These vacuum-based devices often operate in the space-chargelimited regime [12], [13] by extracting microwave power due to an inherent instability of the electron flows in the crossedfield gaps [14].
For B = 0, the critical current density is the SCLCD, which has been studied for different geometries [26], [27], [28], [29], multiple dimensions [30], [31], [32], [33], and nonzero initial velocity [33], [34].The SCLCD (or critical current in the case of the crossed-field case) considering specific effects is often represented as a modification of J CL .Multiple correction factors may be applied to (2) to incorporate the contributions of other physical phenomena, such as an orthogonal magnetic field [23], multiple dimensions, and nonzero initial velocity [33].Introducing a magnetic field orthogonal to the electric field in the gap modifies the maximum emitted current density permissible to J CR , which may be approximated as a crossed-field SCLCD (CF-SCLCD) [7].
If B > B H , in which the electrons execute cycloidal orbits (without space-charge) and return to the cathode, a critical current J CR may be found to mark the transition or collapse from a stationary cycloidal flow to the non-stationary near-Brillouin (nB) flow [3], [6], [23], [35].The nB flow consists of the classical Brillouin flow [3], [16], [36] superimposed by a turbulent background characterized by virtual cathode (VC) oscillation [2], [3], [18], [19].A crossed-field gap becomes more susceptible to nB flow under any slight perturbation, such as applying a small AC voltage across the gap in addition to the DC bias [17], adding an external series resistor [18], or introducing a component of the magnetic field parallel to the electric field (which additionally eliminates magnetic insulation for any B) [19].Because all these conditions cause space-charge to buildup in the gap [18], [19], or VC formation right in front of the cathode [17], the electron trajectories eventually collapse to nB flow.
However, such a transition or collapse to the nB state may only occur when a uniform VC can be formed, such as in 1D [3], [7], [15], [17], [18], [19], [37] or two-dimensions (2D) with a crossed-magnetic field normal to the infinite direction (neglected depth) [23] (e.g., B is in the y-direction, as shown in Fig. 1).This resembles higher dimensional SCLCD problems where a nonuniform VC appears initially in a limited region [30], [33], [38].The critical current density J CR for 1D and 2D planar CFDs was found analytically by assuming a space-charge-limited condition and examined by electrostatic, particle-in-cell (PIC) simulation in the nonrelativistic regime [3], [23].Note that this space-charge limited condition agreed reasonably well with the actual critical current derived separately for 1D [39].The 1D theory was reproduced using variational calculus and extended to cylindrical geometry [7].
For a magnetic field applied parallel to the infinite direction (e.g., B aligns with the z-direction in Fig. 1), our 2D PIC simulations exhibit no complete and sudden collapses of cycloidal flow to an nB state as in 1D.Instead, for J > J CR , the VC may only form in a limited region on the cathode, generating both cycloidal and Brillouin components.In fact, Brillouin components appear even for J in ≪ J CR,1D [3] for a large emission width with the contribution of the Brillouin components increasing with increasing J in .A 3D simulation of electron perturbation induced by a magnetic field tilt also demonstrated no sudden collapse of cycloidal flow [20].
The absence of a transition in electron trajectory modality makes using either the cycloidal or Brillouin model alone to describe the electron beam impractical.Instead, in this work, we quantify the relative contributions of each individual mechanism to characterize the electron trajectories and relative stability.Partially inspired by Slater's orbits that generalize both cycloidal and Brillouin orbits [13], we developed two metrics to assess the Brillouin and cycloidal components in a 2D planar crossed-field gap for various emission widths and injection current densities.We used the analytical solution of both Brillouin and cycloidal flows and 2D PIC simulation results to construct the metrics.We further studied VC formation and how the VC altered these metrics.While neither our metrics nor Slater's orbits describe an oscillating magnetron, they may elucidate the behavior in the oscillatory range [13].Similar to the methods developed for analyzing oscillating magnetrons based on PIC data [40], the present assessments may be applied and extended to other similar devices or physical models, such as magnetically insulated line oscillators [16], two-species Brillouin flow [41], Hall thrusters [42], and Z-pinch [43].
Section II describes the model and PIC simulations for assessing the electron flows in a crossed-field gap.Section III discusses the results.We make concluding remarks in Section IV.

II. MODEL AND SIMULATION A. MODEL PARAMETERS
We consider a crossed-field gap in a 2D planar geometry using XOOPIC (2D Object-Oriented Particle-in-Cell code [44]) with electrostatic and nonrelativistic settings.Fig. 1 shows the simulation geometry and parameters.The cathode was grounded at x = 0 and an anode was biased to V g at x = D with a magnetic field ⃗ B in the −z-direction.For comparison, we used the device parameters based on the Navy Aegis CFA that was used in prior 1D crossedfield studies that determined the 1D critical current density J CR,1D that marks the transition from cycloidal flow to near-Brillouin (nB) flow [2], [3].We fixed the gap distance D = 0.00216 m, magnetic field B = 0.27 T, bias voltage V g = 12 kV, and electron injection energy of 0.5 eV (normal injection with initial velocity v 0 ≈ 0), which gives B/B H = 1.579 [3].We set the length of the cathode L = 8D with dielectric boundaries at y = 0 and y = L.While applying periodic boundary conditions at y = 0 and y = L may resemble a pseudo-cylindrical crossed-field device, we focus on a planar crossed-field device here with dielectric boundaries that drain electrons.Electrons were emitted from y = L to y = L − W , where W is the emission width.We selected emission from this region to ensure that electrons had adequate space (and, concomitantly, time) to complete a cycloidal orbit after emission.We recorded raw data from XOOPIC (such as electron number density and velocity profiles) after the number of macroparticles reached a steady-state with minimal variation (typically < 5% of the steady-state number).Note that applying periodic boundaries would cause the number of macroparticles to increase without bound rather than reach a steady state.The numerical settings, with a square mesh size with dimensions x = y = 5.4 × 10 −5 m and a time step of t = 9 × 10 −14 s, are identical to our previous work [33] to ensure numerical stability [45].We used the analytical Brillouin flow solutions in 1D to evaluate the components of cycloidal, Brillouin, and cycloidal noise from XOOPIC models.We assume that electron flow consists only of these three components.The analytical velocity components for electrons in 1D Brillouin flow in the x-and y-directions are given by [4] and [13] and respectively, where ω c = eB/m e is the electron cyclotron frequency.Note that (4) is identical to the analytical representation of cycloidal flow without space-charge.The number density for Brillouin flow, which is constant, is obtained by , where ω p is the plasma frequency, and rearranging to obtain [4], [13]

B. BRILLOUIN COMPONENTS AND CYCLOIDAL NOISE
Fig. 2 shows the velocity in the x-direction at the p th data point v x (x p ), where x p is the displacement in the x-direction at the p th data point, from XOOPIC compared to the analytical Brillouin flow solution of v x (x) from (3).Fig. 3 compares the velocity in the y-direction v y (x p ) from XOOPIC to the analytical Brillouin flow solution of v y (x) from (4).
Increasing J in causes more particles to exhibit Brillouinlike behavior [i.e., v x x p = 0], as shown in Fig. 2(b), compared to the cycloidal orbit from Fig. 2(a).Although more particles exhibit Brillouin-like behavior, this buildup 11380 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.  of space-charge with no (or little) velocity in the x-direction is insufficient to induce a complete collapse of the cycloidal orbit, as observed in previous 1D simulations for J in > J CR,1D [3] or 2D simulations with ⃗ B in the y-direction [23].Note that assessing Brillouin components based on the analytical solution for Brillouin flow and PIC simulations is approximately equivalent to assessing near-Brillouin (nB) components.Ideal Brillouin flow is used as a theoretical approximation to the nB flow in simulations [35].The Brillouin components that fully satisfy (3)-( 5) cannot be separated from the nB components in 1D or 2D PIC simulations.The Brillouin components always coincide with a VC region (at x ≈ 0) with a number density much higher than n b in (5) both in 1D XPDP1 [2], [3], [46] and 2D XOOPIC [44] in this work.Our XPDP1 results that we report later show that the velocity profile of nB flow in 1D is identical to (3) [2], [3] and (4).
Fig. 3 shows that raising J in also generally increases the cycloidal noise.Because Brillouin and cycloidal flow have identical v y (x) from (4), we cannot distinguish between them based on v y (x) alone.However, we can assess the cycloidal noise in the system by examining the relative difference between the analytical solution v y (x) = −ω c x and the PIC result v y x p , or defining v y x p = v y x p − (−ω c x p ).
Quantifying the relative contribution of Brillouin or cycloidal flow requires first calculating the fraction F of particles as a function of v x (x) and v y (x), respectively, with scaling/normalization using the number density in each case.Appendix summarizes the process for using MATLAB to determine F as a function of either v x (x) or v y (x) from the XOOPIC simulations.
Fig. 4 shows F as a function of v x (x) and v y (x) for various J in with W = 8 D. As J in increases, the variation of F suggests more Brillouin components [with higher fraction F in v x ≈ 0 and v y (x) ≈ 0] and cycloidal noise [higher F in large | v y (x) |] components.Fig. 4(a) exhibits ''Brillouin peaks'' in F for v x ≈ 0 for sufficiently high J in , suggesting stronger ''Brillouin-ness'' in the electron flows.In general, the distributions of v x for various J in in Fig. 4(a) are symmetrical to v x = 0, while the distributions of v y (x) are asymmetrical to v y (x) = 0, as shown in Fig. 4(b).The latter indicates that more electrons have lower speeds, compared to (4), which corresponds to an analytical (or single particle) cycloidal trajectory or Brillouin flow, while traveling along the -y-direction but exhibit a higher maximum displacement in the x-direction in the presence of greater cycloidal noise, as shown in Fig. 3(b).This could be important to magnetron startup [40] according to the BH condition [11].

C. VIRTUAL CATHODE (VC) FORMATION
A VC may appear for sufficiently high J in .Fig. 5 shows the potential φ(x, y) for various J in with W /D = 8.For high J in , the VC first appears where the newly emitted electrons interact with the returning electrons, creating a small spacecharge-limited region, as shown from 0 < y ≲ 0.012 m in Fig. 5(b).From ∼ 0.012m ≲ y < L, there are no returning electrons, minimizing (or eliminating) the spacecharge effect.For higher J in , the VC would nonuniformly cover the entire emission region, as shown in Fig. 5(c).The nonuniformity of the VC formation as J in increases may explain why cycloidal flow does not collapse to nB in 2D cases with B z for J in ≫ J CR,1D = 2.1×10 5 A/m 2 [3] for large W /D, despite the asymmetry of the electron flows caused by the crossed B z in our 2D geometry or in any 3D geometry.Note that we assume J CR,2D,B z ≈ J CR,1D as W /D ≫ 1, where J CR,2D,B z is the 2D critical current density with an applied B z .
This differs from 1D or 2D (with B y ) scenarios in which the returning electrons will always interact with the departing ones.In fact, both 1D and 2D theories considered the interaction of the returning and newly emitted electrons [3], [23].Both J CR,1D and J CR,2D predict VC formation and mark the transition from cycloidal flow to nB flow.This raises the question of how the appearance of the VC would affect the electron flows in the 2D crossed-field gap with B z considered here.
In our XOOPIC models, the 2D critical current density J CR,2D,B z may only be determined by identifying the potential derivative dφ dx = 0 or the electric field E x = 0 at x = 0, which corresponds to the SCLC for zero emission velocity [2].A more practical way is to identify whether E x > 0 at x = 0 by varying J in in a binary search manner [33].The resulting J CR,2D,B z from XOOPIC is the minimum J in that results in VC formation.We discuss the results in the next section.
Unlike our previous studies using XOOPIC for B = 0 [33] with extended regions near the dielectric boundaries, electrons may accumulate at the cell associated with x = 0 and y = 0 where the dielectric boundary (at y = 0) and Dirichlet boundary (at x = 0) intersect for W /D = 8, which is an intrinsic and inevitable numerical issue in XOOPIC for this calculation.This causes E x > 0 at x = 0 and y = 0 even with low J in .Thus, this false VC must be ignored when calculating J CR,2D,B z from XOOPIC.In the next section, we will demonstrate that this does not significantly influence our assessment.

III. RESULTS AND DISCUSSION
To compare the results in Fig. 4 and other emission widths W , we further introduce two quantities C B or C T that serve as metrics, or figures of merit, for each curve to assess the Brillouin and cycloidal noise components, respectively.Specifically, we considered and where N q is the number of the intervals (bins) of v x (x q ) or v y x q .Note that ⟩ to evaluate the Brillouin components with smaller/larger values indicating greater Brillouin/cycloidal contributions.We define C B as the reciprocal of this value to make it directly proportional to the ''Brillouin-ness'' of the flow.For C T , since v y (x) = −ω c x for both Brillouin and cycloidal flow, a larger overall difference from this value [i.e., a larger ⟨( v y (x q )) 2  ⟩] indicates greater cycloidal noise.Equation (7) indicates that C T is directly proportional to the cycloidal noise.Fig. 6 shows (a) C B and (b) C T for W /D =0.5, 1, 2, 4, and 8 as a function of J in from XOOPIC and XPDP1 [2], [3], [46] simulations.We also examined the time-dependent behavior of C B and C T up to 8 ns.Both quantities reached steady states, as indicated by the number of macroparticles becoming stable with minimal variation.
For 1D XPDP1 simulations, C B and C T models the sudden change for J in = J CR,1D , corresponding to the transition from a stable cycloidal flow to the nB state, while the 2D results do not dramatically change behavior for J in ≈ J CR,2D .Specifically, C B increases gradually as J in increases for 2D models.Although the methods of data processing of raw PIC data for XPDP1 and XOOPIC are similar, the two metrics from 1D or 2D PIC do not seem directly comparable due to the difference behavior at this transition.
Fig. 6(a) shows that for most 2D cases, C B decreases with decreasing W for a given J in .This resembles the results in 2D nonmagnetic cases [30], [33], [38], where the SCLCD for W D ⪅ 1 is much higher than those determined for either large W D or 1D.For the crossed-field case, this occurs 11382 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.because VC formation for a smaller W /D requires a higher J in , which increases the limiting current (B < B H ) or reduces the Brillouin components (B > B H ). Since Brillouin flow also arises due to VC formation, or space-charge buildup at the cathode [3], [17], [18], [19], similar to SCLCD, we anticipate similar behavior.
For smaller W , Fig. 6(b) shows that C T also peaks and decreases at higher J in , similar to VC formation.Thus, we conjecture that VC formation may affect the cycloidal noise, or C T .Fig. 7 compares J CR,2D from XOOPIC to the injection current density J in,p associated with the maximum value of the metric C T for each W D considered.We obtained J CR,2D from XOOPIC by varying J in until reaching the minimum value that caused VC formation such that further iterations induced negligible error [33].When determining J in,p associated with the maximum C T , we only obtained J in,p from our tested values of J in without iterations.
We may also plot 2D semi-empirical and analytical J CR,2D [23], although they should be applicable only for B y .We assume they both should provide reasonable estimates for B z , at least for large W /D with zero initial velocity, similar to B < B H [23].The semi-empirical form of J CR,2D , similar to the semi-empirical 2D uniform SCLC for v 0 ̸ = 0 and B = 0 [33], is given by where the field correction factor [3] is given by and the geometry correction factor [30] is given by Note that the exact solution for the 1D crossed-field case is J CR,1D = J CL p B, V g , D [3].The semi-empirical form in ( 8) is obtained simply by multiplying J CL from (2) by the two correction factors from ( 9) and (10).This is based on a conjuncture that additional effects that contribute to SCLC may be considered as multiplicative correction factors to J CL for rapid estimates, which we have shown is reasonable for nonmagnetic cases with sufficiently low v 0 [33].Fig. 7 shows that such a semi-empirical form gives similar values compared to the theoretical J CR,2D [23].However, at smaller W /D, the semi-empirical relationship deviates from the theory, similar to 2D uniform SCLCD with nonzero v 0 [33].TABLE 1.Estimated C B associated with J CR,2D from XOOPIC for each W /D using linear interpolation from the results in Fig. 6(a).
Both the semi-empirical and theoretical J CR,2D disagree with J CR,2D obtained from XOOPIC, even for relatively large W /D in which the returning electrons will interact with departing ones.1D and 2D theories typically assume that the returning and departing electrons interact [2], [3], [23].The J CR,2D from XOOPIC follows a similar trend as (and agrees reasonably well with) J in,p obtained using the maximum value of C T .Thus, we speculate that the cycloidal noise reaches a maximum and starts to decrease as the VC starts to form and expand along the cathode, as shown in Fig. 7. Fitting J CR,2D from XOOPIC and J in,p using J = a W D b gives a = 3.44×10 5 A/m 2 and b = −0.5679for J CR,2D from XOOPIC (with R 2 = 0.9814) and a = 4.14×10 5 A/m 2 and b = −0.5418for J in,p (with R 2 = 0.9749) as the best-fit parameters.
In addition, since J CR,2D marks the appearance of a VC for each W D, we may estimate C B by applying linear interpolation to the results in Fig. 6(a).Table 1 shows similar C B (∼1.5×10 −14 s 2 m −2 ) associated with J CR,2D from XOOPIC for each W /D.This indicates that the Brillouin components are similar for each W upon VC formation, although J CR,2D varies for different W /D. It also shows the charge accumulation issue that arises due to the intersection of the planes for  the Dirichlet and Neumann boundary conditions for W /D = 8 does not appreciably affect the evaluation of J CR,2D in XOOPIC.Furthermore, for J in > J CR,2D , the rate of change of C B and C T with respect to J in eventually becomes essentially the same (''similar slope'') for each emission width after VC expansion.Although we have shown that J CR,2D may not be used to describe the electron flows in 2D crossedfield gaps where there is not binary behavior (i.e., a direct transition from cycloidal flow to nB flow upon exceeding a threshold critical current density), the formation of the VC associated with J CR,2D may still significantly affect the variations of Brillouin and cycloidal noise components.

IV. CONCLUSION
In 2D crossed-field gaps (CFGs), while the critical current density still characterizes the appearance of a VC, it no longer signifies the transition from stationary cycloidal flow to the nB state observed in 1D.Instead, Brillouin components coexist with cycloidal flows and noise, even for J in ≪ J CR,2D .
In this study, we developed two metrics for evaluating the components of near-Brillouin and cycloidal noise components in a 2D planar CFG with various W D and J in .
The metrics rely on data obtained from 2D PIC simulations and analytical solutions for cycloidal and Brillouin flow.For smaller W D, Brillouin flow contributes less to the beam for a given J in .Cycloidal noise initially increases with increasing J in and VC formation before reaching a peak and decreasing.Furthermore, the Brillouin components behave similarly across W /D upon VC formation, although J CR,2D varies with W /D. The assessments potentially connect the newly developed metrics and the traditionally used critical current density for VC formation and transition to nB state for B > B H in 1D [2], [3], [7], [17], [19] or 2D with B y [23].Moreover, the presented methodology of assessing the 11384 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

FIGURE 1 .
FIGURE 1. Schematic of a representative magnetically insulated crossed-field gap (magnetic field B >B H , where B H is the Hull cutoff magnetic field) used in the simulations with cycloidal electron trajectories (without space-charge).The grounded cathode from which electrons are emitted is at x = 0 and the anode at x = D, is held at a potential V g .The dielectric boundaries, where the electrons are drained off, were set at y = 0 and y = L.

FIGURE 2 .
FIGURE 2. Electron velocity in the x-direction v x (x) from XOOPIC with injection current density J of (a) 5.79 × 10 4 A/m 2 (J ≪ J CR,1D , where J CR,1D is the 1D critical current density) and (b) 2.315 × 10 5 A/m 2 (J > J CR,1D ) compared to the analytical Brillouin flow velocity profile v x (x) = 0 in red.

FIGURE 4 .
FIGURE 4. The fraction of particles F (dimensionless) for a given (a) v x (x p ) and (b) v y (x p ) for various injection current density with emission width W = 8D.

FIGURE 5 .
FIGURE 5. Potential φ(x, y ) with W /D = 8 and injection current densities of (a) 5.79×10 4 A/m 2 without VC formation, (b) 1.74×10 5 A/m 2 with a VC from 0 < y 0.012 m where the newly emitted electrons interact with the returning electrons, and (c) 5.79×10 5 A/m 2 with a VC covering the entire emission region.

FIGURE 6 .
FIGURE 6. Brillouin components and cycloidal noise assessments: (a) C B and (b) C T as a function of injection current density J in for W = 0.5D, D, 2D, 4D, 8D and 1D models from XPDP1.

FIGURE 7 .
FIGURE 7. Comparison of the 2D critical current density J CR,2D from the analytical form [23], the semi-empirical equation (8), PIC, and the injection current density J in,p associated with the maximum value of C T for assessing cycloidal noise as shown in Fig. 5(b).We fit J CR,2D from XOOPIC and J in,p using J = a W D b , where a and b are fitting parameters.We obtained a = 3.44×10 5 A/m 2 and b = −0.5679for J CR,2D from XOOPIC (with R 2 = 0.9814) and a = 4.14×10 5 A/m 2 and b = −0.5418for J in,p (with R 2 = 0.9749) as the best-fit parameters.