Coordinate System Invariant Formulation of Fractional‐Dimensional Child‐Langmuir Law for a Rough Cathode

This work presents an exact closed‐form analytical model of the space charge limited (SCL) current density for planar, cylindrical, and spherical geometries in fractional dimensional spaces (Fα) where α represents the fractional dimensions having a range 0 < α ⩽ 1. Decreasing value of α shows an increasing degree of surface roughness. This analytical model approaches Langmuir–Blodgett equations for sufficiently close anode (Ra) and cathode (Rc) radii for integer dimensions (α = 1). To calculate SCL current density for any geometry, variational calculus (VC) is used to derive the differential equations in fractional (α) dimensions. The effect of electrode radius ratios on space charge limited current is investigated here considering the surface roughness.


Introduction
Maximum current density across a gap is a major concern in modern devices that is, high-power microwave circuits, accelerators, and micro-electronics. The Child Langmuir (CL) law is the first law that describes space charge limited emission (SCLE) in DOI: 10.1002/apxr.202200120 a vacuum. Maximum current density through a 1D planar surface is described in terms of voltage and distance between the gap by the CL Law. [1,2] The CL law has a wide variety of applications and is the most important law for modeling devices that is, diodes, beams in electron guns, etc. hence, making this area an important regime of research. On the other hand, CL law makes a number of assumptions, which are then addressed by various researchers depending on the device's specifications. Different research has been done on SCLE in different systems with different geometries and aspects. One of the most well-known utilities of this law is to achieve maximum current using a unique cathode structure in a device named Pierce Gun. [3] Moreover, in ref. [4], the addition of positive ions is done in different devices to neutralize the space charge effect which is crucial in different modern device implications. The SCLE also have different impacts in cathode fall. [5] Further, different extensions have been done in this law for different geometries and different materials leading to the research of finding SCL current in various devices having material properties such as solid, insulators, and semi-conductors [6,7] or involving different components such as cones, nano-tube structures, or sharp edges. [8,9] It also involves different applications in different processes as photoelectric effect [10] and quantum tunneling phenomenon. [11] The 2D CL law [12] showed that maximum current density is far greater than that of classical CL law and further 3D analysis is reported in ref. [13].
Advanced theories have extended this law for different conditions such as injecting velocity, electron limits, and different scaling of geometries. [14] Extension of CL law for spherical geometries has also been reported in ref. [15] which leads to more extensions that is, for cylindrical surfaces. [16] The 2D surface analysis of CL law has also been done which involves 2D Cartesian and cylindrical analysis. [17,18] Moreover, besides the cumbersome solutions of SCLC, there is a requirement of approaches to generalize the expression of it for different geometries as well as considering the roughness of surfaces, which has a crucial effect on nano-level devices. [19] A better approach has been used in ref. [20] where maximum current density has been calculated for fractional dimensions. By applying conformal mapping and VC, a mathematical relationship between vacuum potential and www.advancedsciencenews.com www.advphysicsres.com SCL potential for multi-dimensional geometry of a diode has also been derived and used to determine an exact solution for SCLC density (SCLCD) from a finite emitter. [21] Due to the contemporary needs in studies of modern devices that is, nano-scale devices, researchers are interested in deriving generalized SCL current expressions about applied voltage for cylindrical and spherical geometries as well. Many researchers have proposed methods for calculating maximum current density in cylindrical and spherical geometries by different approximations that is, Langmuir Blodgett (LB) equations analysis using numerical or analytical methods, etc. Some of the past research involves cumbersome calculations that are, utilizing the power series with different approximations hence limiting the size of electrodes in the system. [10,22] Conformal mapping has been used to determine SCL current densities for 1D concentric cylinders, and for many other complicated geometries based on geometric arguments. [23] Lie-point symmetries have also been used to derive exact analytical equations for SCL current density for 1D spherical, tip-to-tip (t-t), tip-to-plate (t-p), and cylindrical diodes. [24] In recent works, [25][26][27] variational calculus (VC) techniques are suggested to calculate maximum current density using exact closed-form expression of CL law which is a big step further as compared to the numerical solutions.
In this study, solutions of maximum current density have been derived for a rough cathode using VC, and exact closed-form solutions have been determined for planar, cylindrical, and spherical geometries. For imperfect and rough boundary conditions, Euler Lagrange (EL) equation is proposed to be used, to determine analytical equations for the closely placed anode and cathode radii.
The rest of the paper is organized as follows: In Section 2, a mathematical formulation of CL law in Cartesian, cylindrical, and spherical coordinates is considered for a rough cathode. The main results and discussions have been provided in Section 3. Also, the analysis has been done by keeping both the cathode and anode radii constant. Finally, the main contributions are summarized in Section 4.

Cartesian Coordinate System
An infinite 1D parallel plate diode in fractional-dimensional space(F ) where 0 < ⩽ 1, have Poisson's equation defined as [25,28] where ϵ o is free space permittivity and is defined as charge density, and ∇ 2 in fractional-dimension is defined as with c( , x) = ∕2 Γ( ∕2) |x| −1 . The law of conservation of energy for electron (e) with mass (m) and velocity (v) is defined as According to equation of continuity, On combining Equations (1) and (3), current density is defined as The total current in an area A is given as I=JA. Hence complete expression for current [22] is where ds is differential length between electrodes while f = where Π n j=1 (x j , j ) is combination of all sections of rough surface area in diode. By using Euler Lagrange equation in VC and minimizing energy, the expression of voltage in fractional dimension is defined as By applying the fractional dimensional Laplacian equation, Equation (7) can be rewritten as For 1D diode, Equation (8) is solved to get the differential equation as This equation is modified Emden's Fowler equation and further solving the system for boundary conditions of V(0) = 0 and V(L) = V o , given V o and L are applied voltage and distance between the two electrodes results in Combining Equations (4) and (10), space charge limited current (SCLC) in Cartesian coordinate is defined as Equation (11) is the CL law for fractional dimensional space which reduces to integer dimensional CL law J 1 =

Cylindrical Coordinate System
Consider a 1D rough surface cylindrical diode. By substituting both the cylindrical fractional Laplacian and gradient in Equation (7), results in Equation (12) is further modified as Equation (16)

is modified Emden's Fowler equation and provides the voltage at boundary conditions of
where R a and R c are anode and cathode radii. Hence from ref. [29], two solutions of the equation are derived depending on the values of c/b and d/b. Case I: In the first case with = 1 and ϕ = 0, Equation (16) is derived as On solving Equation (17), the voltage at boundary conditions is derived as Expression for Laplacian is derived by substituting Equation (18) in Equation (7) that results in Further, combining Equations (4) and (19), fractional SCLC is derived as Case II: In this case, 0 < < 1, and general expression for a given range of is derived by applying boundary conditions defined above. The fractional generalized expression for voltage is derived as This expression is further used to evaluate SCL current, derived as where the values of b, c, and d are calculated as 1, −1, and 1, respectively. On substituting ϕ = 0, voltage is calculated as and 1D SCL current for a rough cathode in the cylindrical system is derived as Hence, analyzing the equations derived for both cases of cylindrical CL law, the equation for Case I when = 1 is exactly similar to that of Darr and Garner's cylindrical CL law [25] while for Case II when 0 < < 1, plots are derived which shows the relationship between SCL current and voltage applied for vacuum diode. Moreover, it is prominent in the plots that surface roughness is directly related to SCL current and inversely with value of .

Spherical Coordinate System
Consider a rough surface cathode with 1D variation that is, only along r. Start with the conversion from Cartesian to spherical coordinates to find Laplacian and gradient in fractional dimensions for spherical coordinates. Utilizing the conversion formulas from Cartesian to spherical coordinates, and substituting both the Laplacian and gradient in Equation (7), the equation is derived as where, t = sin 2 cos 2 (sin cos ) 2( −1) + sin 2 sin 2 (sin sin ) 2( 1) + cos 2 (cos ) 2( −1) The more generalized fractional expression is defined as Again the above equation is modified Emden's Fowler equation and boundary conditions are defined as V(R c ) = 0 and V(R a ) = V ( g) , where R a and R c are anode and cathode radii, thus expression for voltage is derived as where, Hence, the final expression for voltage is derived as This expression is further used to evaluate SCLC, derived as The values of t, f, and g are calculated as 1, −2, and 1 at = 1, respectively. On substituting these values in Equation (32), the expression for voltage in fractional dimensions is derived as Solving the equation further gives a final expression for voltage, which is the desired result as that of Darr and Garner's approach [25] Further on substituting the above expression in Equation (33) and at r = R c surface charge current density is given as where, The expression derived for spherical coordinates is comparable with the equations derived in refs. [20,25] that is, with increasing the value of (decreasing the roughness of cathode) the value of SCLC increases. Figure 1 shows the analysis of variation in SCLC in relation to the radius ratio of electrodes. Considering the same approach as of Allen's, [25] results are validated for rough surface electrodes in Cartesian coordinates, that is, planar surface rough surface cathodes. Figure 1a shows the variation in SCLC when the inner radius is kept constant at 1 cm either its cathode or anode. The SCLC is maximum when the radii of both electrodes are equal. Further increasing the radius of cathode as compared to anode, the SCLC starts decreasing, hence validating the equation for CL law in fractional dimensions. While the radius of the cathode is constant in Figure 1b, it is determined that when the cathode is the outside electrode, the fluctuation in current values is relatively minimal in comparison to when the cathode is the inner electrode. Furthermore, when the value of is examined, an increase in the value of results in a decrease in the surface roughness of electrodes. As a result, as surface roughness decreases, so does SCLC. Further investigation into the effects of the radius ratiō a = R c ∕R a reveals that when the radius ratio increases, electrons emitted from the cathode begin to accumulate at the anode and a crowd of electrons is produced at the anode, increasing the electric field, which results in a reduction in SCLC.

Results and Discussion
Using rough surface electrodes with cylindrical symmetry, the effect of roughness and radius ratio on SCLC has been investigated in Figure 2. Variation in cathode and anode radii has an effect on space emissions. As the radius ratio is increased, space charge accumulation begins to reduce the current. Additionally, as electrode radii become closer to one another, the SCLC reaches its maximum value, which then decreases as the radius ratio increases or decreases. The results of an investigation into the effect of voltages on SCLC in both cases reveal that voltage increases the current by a factor of 1.5 in both cases. As a result of this investigation, it is discovered that the CL law holds true for fractional cylindrical electrode surfaces.
In vacuum diodes with spherical electrodes, the increment in sphere radii has similar effect on space charge emissions as in the case of rough surface planar electrodes. Thus increasing the radii of the sphere decreases the SCLC which is completely in agreement with Darr and Garner's approach [25] for spherical electrode smooth diodes as shown in Figure 3. An analysis is done for both cases when the inner electrode, either cathode or anode is kept constant. For both cases, the results achieved are  completely comparable with the equations derived using variational calculus (VC). Analyzing the equations derived for spherical electrodes in fractional dimensions, CL law is completely validated in this scenario and it is found that increasing the value of reduces the surface roughness, and reduction in surface roughness leads to the reduction in SCLC. While the relationship between voltage and current remains the same as the above two coordinates, thus validating the CL law for spherical electrodes in fractional dimensions. Finally, it is worth mentioning that analysis of various reported experiments has shown that non-ideal surface conditions can be modeled with fractional models where fractional parameter ( ) may take values less than unity. [20]

Summary
This study employs variational calculus to derive exact closedform analytical solutions to SCL current problems in fractional dimensions for Cartesian, cylindrical, and spherical systems. The space-fractional Laplacian operator validates continuity for all three coordinate systems, which trivially takes care of the fractional dependence on the radial variable r when used in the continuity equation (see Appendix). The proposed fractional dimension models were rigorously analyzed numerically, the findings of which are included in the appropriate section. A careful examination finds that the suggested study is highly consistent with prior published work in the same field. Additionally, the derivations are completely comparable to the closed-form solutions of the fractionally generalized CL law. The proposed methodology, strategy, and final model are expected to be more computationally efficient than standard PIC simulations, allowing for quick solution screening. A complete analysis of the radius ratio on SCLE and the effect of varying the radius of electrodes have been provided in this study which helps in designing the different devices.
In the future, the VC approach can be used for multidimensional rough surfaces systems [30,31] and for systems with different mediums besides vacuum. Additionally, the proposed method has theoretical implications for quantum CL and transitions between SCLE and field emission in a vacuum. [32][33][34][35] The proposed fractional VC approach could be applied to t-t and t-p geometries by carefully observing the impact of electrode surface roughness on electrode work function. [36] (

A.2 Cylindrical Coordinate System
The divergence of the vector field in fractional dimensions for cylindrical coordinate system has the form The space charge limited current for the cylindrical diode in fractional dimensions is derived as Substituting value of J in Equation (A8)

A.3 Spherical Coordinate System
The divergence of the space charge current density in fractional dimensions for spherical coordinate system has the form (A15) The SCLC for spherical diode in fractional dimensions is derived as