A coordinate system invariant formulation for space-charge limited current with nonzero injection velocity

Halpern, Jacob; Darr, Adam M.; Harsha, N. R. Sree; Garner, Allen L.

doi:10.1088/1361-6595/ac89a9

Cited by 10 publications

(2 citation statements)

References 68 publications

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“…With decreasing (i.e., ), . Figure 6 b shows as a function of , which is the standard scaling for CL with nonzero for planar 49 – 51 and nonplanar 53 , 54 diodes. As , ; as , continues to increase.…”

Section: Resultsmentioning

confidence: 93%

“…Thus, the results reported here, particularly for the fourth-order nexus between RLD, FN, GMG, and GCL, elucidate the contributions of the emission mechanisms (RLD and FN) and limits (GMG and GCL) for thermionic emitters in nano- and microscale devices that may undergo gas breakdown. Such theories may ultimately be extended to include additional emission mechanisms, such as photoemission 34 , or nonplanar geometries, which have been an ongoing area of study for both zero 59 and nonzero 53 , 54 injection velocity. Meadors and Poirier studied how to use a laser to heat a cathode to induce thermionic emission in vacuum and at atmospheric pressure without electromagnetic interference 60 .…”

Section: Discussionmentioning

confidence: 99%

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The transition from field emission to collisional space-charge limited current with nonzero initial velocity

Breen,

Loveless,

Darr

et al. 2023

Sci Rep

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Multiple electron emission mechanisms often contribute in electron devices, motivating theoretical studies characterizing the transitions between them. Previous studies unified thermionic and field emission, defined by the Richardson-Laue-Dushman (RLD) and Fowler–Nordheim (FN) equations, respectively, with the Child-Langmuir (CL) law for vacuum space-charge limited current (SCLC); another study unified FN and CL with the Mott-Gurney (MG) law for collisional SCLC. However, thermionic emission, which introduces a nonzero injection velocity, may also occur in gas, motivating this analysis to unify RLD, FN, CL, and MG. We exactly calculate the current density as a function of applied voltage over a range of injection velocity (i.e., temperature), mobility, and gap distance. This exact solution approaches RLD, FN, and generalized CL (GCL) and MG (GMG) for nonzero injection velocity under appropriate limits. For nonzero initial velocity, GMG approaches zero for sufficiently small applied voltage and mobility, making these gaps always space-charge limited by either GMG at low voltage or GCL at high voltage. The third-order nexus between FN, GMG, and GCL changes negligibly from the zero initial velocity calculation over ten orders of magnitude of applied voltage. These results provide a closed form solution for GMG and guidance on thermionic emission in a collisional gap.

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Section: Resultsmentioning

confidence: 93%

Section: Discussionmentioning

confidence: 99%

The transition from field emission to collisional space-charge limited current with nonzero initial velocity

Breen,

Loveless,

Darr

et al. 2023

Sci Rep

View full text Add to dashboard Cite

show abstract

Electron trajectories in a collisional crossed-field gap

Garner

Allison

Breen

et al. 2023

Applied Physics Letters

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The Hull cutoff represents the maximum magnetic field in a vacuum crossed-field gap (CFG) such that an electron emitted from the cathode reaches the anode. Prior studies demonstrated that introducing ions into a CFG always causes increased excursion of electrons toward the anode. In this paper, we assess a collisional CFG by incorporating collision frequency into the electron force law. The theoretical electron trajectories agree well with a one-dimensional particle-in-cell simulation and demonstrate that emitted electrons always cross a collisional CFG. We derive a modified Hull cutoff condition for a collisional CFG corresponding to an electron reaching the anode with zero velocity in the direction of the electric field. Rather than representing the threshold for magnetic insulation, this condition gives the maximum magnetic field and maximum collision frequency for which an electron reaches the anode without turning around; higher magnetic fields and/or collision frequencies cause the electron to turn around before crossing the gap. Further increasing either quantity causes the electron to change direction more frequently as it crosses the gap, noticeably increasing the transit time with each change in electron direction. In the limit of high collision frequency, the electron velocity across the gap approaches a constant, meaning that electrons will reach the anode at nonzero velocity. The transit time above this condition increases smoothly and monotonically with increasing magnetic field or collision frequency. These results elucidate the implications of collisions on magnetic insulation for future assessments of the limiting current in a collisional CFG.

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Uniform space-charge-limited current for a two-dimensional planar emitter with nonzero monoenergetic initial velocity

Zhu,

Sree Harsha,

Garner

2023

Journal of Applied Physics

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While characterizing space-charge-limited current (SCLC) is important for numerous applications, no analytical solutions for SCLC with monoenergetic initial velocity exist for two-dimensional (2D) geometries. Here, we derive approximate closed-form solutions for uniform SCLC with monoenergetic emission of electrons in a 2D planar diode, where emission is restricted to a long patch of width W for electrodes separated by a distance D. We also derive a semiempirical approach for estimating the SCLC for these cases by treating the geometric and velocity correction factors as multiplicative corrections to the SCLC for a one-dimensional vacuum diode given by the Child–Langmuir (CL) law. We show that the SCLC for a finite patch with nonzero velocity can exceed the CL law by three orders of magnitude. The theoretically calculated SCLCs for various emission widths and initial velocities in the 2D diode agree well with particle-in-cell simulations using the over-injection method in XOOPIC; they agree with the semiempirical relationship for lower initial velocities. In the limit of high initial velocity, the geometry and velocity corrections to the CL law cannot be decoupled, invalidating the assumption of the semiempirical approach and causing it to diverge from the theoretical solution and XOOPIC simulations. These results provide valuable estimates for determining the onset of virtual cathode formation for photocathodes and thermionic cathodes, which operate in the over-injection regime to avoid beam quality degradation.

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A coordinate system invariant formulation for space-charge limited current with nonzero injection velocity

Cited by 10 publications

References 68 publications

The transition from field emission to collisional space-charge limited current with nonzero initial velocity

The transition from field emission to collisional space-charge limited current with nonzero initial velocity

Electron trajectories in a collisional crossed-field gap

Uniform space-charge-limited current for a two-dimensional planar emitter with nonzero monoenergetic initial velocity

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