Predicting space-charge affected field emission current from curved tips

Field emission refers to the process by which electrons tunnel out from the surface of a conductor on application of a strong electric fieldFN ; Nordheim ; MG ; forbes2006 . The height and width of the tunneling potential barrier depends on the strength of the electric field. As a consequence, the field-emission current depends sensitively on the local electric field on the emitter surface with a small change in field resulting in a large change in emitted current.

The presence of field-emission electrons in a diode can itself be a cause for change in the local field on the emitter surfacestern29 ; ivey49 ; barbour53 ; vanVeen ; jensen97a ; jensen97b ; forbes2008 . If the applied macroscopic field $E_{0}$ is large enough to cause sufficient electron emission, the negative charge cloud lowers the magnitude of the local field on the emitter surface, thereby leading to a decrease in the field emission current. It may also happen that the local field becomes zero and emission stops altogether until the space charge moves away from the cathode and is eventually lost from the diode. Thus, the field emission current in a diode can be oscillatory initially till the local field at the emitter surface saturates with time and a steady-state prevails feng2006 ; jensen2015 ; torfason2015 .

Attempts to determinestern29 ; ivey49 ; barbour53 ; forbes2008 the steady-state cathode electric field and the emission current density for planar emitters have been made since the basic formulation of field emission by Fowler and NordheimFN (FN) in 1928 and subsequent corrections and approximations to the expression for the field emission current density for conductors MG ; forbes2006 . The connection between space charge and the current density is established by expressing the charge density $\rho=J_{P}/v$ where $J_{P}$ is the steady-state current density and $v$ the speedpuri2004 . The speed $v$ can be further expressed in terms of the potential $V$ using energy conservation. Thus, the Poisson equation can be expressed in 1-dimension as

In a planar situation therefore, space charge does affect the field emission current. If we denote the electrostatic field in the absence of any charge by $E_{L}$ and the saturated field (after field emission has continued for a few transit times) by $E_{P}$, a possible measurealternate ; db_scl of the severity of space charge effect may be taken to be the field reduction factorforbes2008 , $\vartheta=E_{P}/E_{L}$. The subscript $L$ and $P$ refer to the Laplace and Poisson equations respectively forbes2008 . Clearly $0\leq\vartheta\leq 1$ with $\vartheta\simeq 1$ denoting the lack of significant space charge effect and $\vartheta=0$ being the classical space charge limit child ; langmuir ; langmuir23 ; db_scl where $E_{P}=0$ and $J_{P}=J_{\text{CL}}$ where $J_{\text{CL}}=(4/9\kappa)V_{g}^{3/2}/D^{2}$ is the Child-Langmuir current.

The dependence of the space-charge affected current $J_{P}$ on the field reduction factor $\vartheta$ can be simplified by re-writing Eq. (2) in terms of the dimensionless quantity $\xi=\kappa J_{P}V_{g}^{1/2}/E_{L}^{2}$ where $V_{g}/D=E_{L}$. The normalization of $J_{P}$ in effect is with respect to the planar Child-Langmuir current and at the space charge limit $\vartheta=0$, $\xi$ assumes the value $\xi_{\text{CL}}=4/9$. Thus, Eq. (2) reduces to

It is difficult to extend the relation directly to the case of curved emitters since $J_{P}$ is no longer uniform. However, a plausible phenomenological relation may be arrived at by recalling a recent result on the space charge limited current for axially symmetric curved emitters. It states that the space charge limited current $I_{\text{SCL}}\approx\pi b^{2}\gamma_{a}J_{\text{CL}}$ where $b$ is the radius of the base of the emitter and $\gamma_{a}$ is the apex field enhancement factor (a constant)db_universal . Thus, it is natural to define the scaled field and current density as

While Eq. (8) together with Eqns. (5-7) constitute a major step in dealing with axially symmetric curved emitters, it is nevertheless an ad-hoc extension of the planar model. The free parameter $\tilde{\omega}$ is a reflection of this approach. Alternately, if $\tilde{\omega}$ is set to unity, the factor $g$ which defines the average current density must be obtained from a fit to numerical results. There is thus sufficient scope and motivation to build an alternate formalism which suits both curved emitters and planar emitters with a finite active area.

A variety of alternate approaches have also been used to study space charge effects on field emission jensen97a ; jensen97b ; lin2007 ; rokhlenko2010 ; jensen2010 ; rokhlenko2013 ; jensen2014 ; jensen2015 . Some of these, based on transit time, are able to reproduce features of the time variation of the cathode field and the steady-state that follows, especially for planar emitters.

The field reduction factor $\tilde{\vartheta}$, which is a measure of space charge severity, may depend on several factors for a planar emitting patch or a curved emitter apex. In a molecular dynamics simulationtorfason2015 , it was found that keeping other features of a parallel plate diode invariant and varying only the size of the planar emitting patch, the value of $\tilde{\vartheta}$ decreases as the size of the patch is increased. Thus, two emitters with emitting areas ${\cal A}_{1}$ and ${\cal A}_{2}$ ($>{\cal A}_{1}$) but having the same $E_{L}$ and transit time $T_{\text{tr}}$, will be affected by space charge differently with the smaller one being relatively less affected torfason2015 . The severity of space charge may also depend on the transit time $T_{\text{tr}}$, the vacuum field $E_{L}$ and several other factors. For an axially symmetric curved emitters, there are very few studies torfason2016 ; rk2021 . The dependence of $\tilde{\vartheta}$ on the transit time $T_{\text{tr}}$ and the vacuum field $E_{L}$ are expected to persist for curved emitters.

The present communication deals with a semi-analytical model, one that naturally accommodates curved emitters and can provide the time variation of the apex field and emitted current and estimates of $E_{P}$ and $J_{P}$ without much computational effort. It aims to predict some of the results reported earlier. While it is not intended as a substitute for PIC methods, it can be used as a design tool to scan the parameter space since a full 3-dimensional PIC modelling may be extremely time consuming and expensive.

We shall first outline the model in section II and state the various approximations that may be used to improve the prediction. This will be followed by a comparison with a published MD simulation for planar geometry and our own PIC simulations using PASUPATdb_scl ; db_multiscale ; rk2021 .

In order to improve the predictive power of the model, it is important to approximate the anode field $E_{A}(t)$. At $t=0$, when the field at the emitter is $\gamma_{a}E_{0}$, the anode field is $E_{0}$. We are also aware from planar space charge limited flows that when the field at the cathode is zero (the limiting case), the anode field assumes the value $4E_{0}/3$. A simple linear interpolation between these two points $(\gamma_{a}E_{0},E_{0})$ and $(0,4E_{0}/3)$ on the $(E_{a},E_{A})$ plane, gives us the relation

If we consider the anode area ${\cal A}(t)$ to be invariant in time and assume its vacuum value, an application of Gauss’s law with only the anode-field correction leads us to the equation

At the next level, we can also introduce an anode area approximation using the parallel-plate diode as a guide. Before emission starts, the flux tube maps equal area at the cathode and anode in a planar diode while in case of a curved emitter, the flux tube maps an area of the cathode that is $\gamma_{a}$ times more at the anode (see Eq. 23). Thus, when the field at the emitter apex is $E_{a}(0)$, the area at anode is $\gamma_{a}{\cal A}_{C}$.

The planar Child-Langmuir law can again be used to obtain a second point. Numerical and analytical results show that if the emitting patch is finite, the Child-Langmuir current density can be expressed as $J_{\text{CL}}(1+\alpha D/W)$ where $W$ represents the size of the emitting patch and $\alpha$ is a constant that depends on the shape of the patch. Thus, for a circular patch, $\alpha=1/4$ while $W$ is its radius. The relevance of the factor $(1+\alpha D/W)$ in the planar case with finite emission area (see Fig. 2) can be understood by considering the Gaussian surface as a the flux tube extending from the cathode having an area ${\cal A}_{C}$, to the anode having an area ${\cal A}_{A}$. On applying Gauss’s law and assuming that the cathode field is zero, we have $-E_{A}{\cal A}_{A}=-Q/\epsilon_{0}$ where $Q=I_{\text{CL}}T_{\text{tr}}=J_{\text{CL}}(1+\alpha D/W){\cal A}_{C}T_{\text{tr}}$. Assuming $E_{A}=4E_{0}/3$ and $T_{\text{tr}}=D/v_{av}$ where the average speed $v_{av}=v_{max}/3=\sqrt{2eV_{g}/m}/3$ as in the infinite parallel plate case, it follows that ${\cal A}_{A}=(1+\alpha D/W){\cal A}_{C}$. In other words, the area of the emitting patch at the cathode is mapped to a patch at the anode that is larger by a factor $(1+\alpha D/W)$. When applied to a curved emitter having an effective emitting area ${\cal A}_{C}$, the area at the anode is $\gamma_{a}{\cal A}_{C}(1+\alpha D/W)$ when the field at the apex is zero.

A linear interpolation between these two points gives

An application of Gauss’s law with both the anode field and area corrections leads to the equation

Eq. (28), Eq. (32) and (35) along with Eq. (36) provide successively better approximations for determining the steady state electric field $E_{P}=E_{a}(\infty)$ at the emitter apex. Note that the value of $\alpha$ is not known with any accuracy and may need to be determined by fitting Particle-in-Cell or Molecular Dynamics data, especially for curved emitters. Nevertheless, it is expected to provide a fast approximate determination of the space charge affected field emission current.

The model presented here is also applicable to a cluster of emitters or a large area field emitter (LAFE). The hybrid model proposed recentlydb_rudra1 ; rudra_db ; db_rudra2 ; anodeprox ; db_hybrid can be used to determine the apex field enhancement factor of individual emitters in the LAFE before emission starts, and so long as two emitters are not too close to each other for mutual space charge effects to kick in, the formalism presented here can be used for each emitter.

A first test of the model is the extensive Molecular Dynamics (MD) data reported in Ref. [torfason2015, ] for a square emitting patch of side length $L$ in a planar diode geometry. The gap between the anode and cathode plates is $1000$nm, the potential difference $V_{g}=2$kV while $L$ varies from $50$nm to $2500$nm. The emission current density follows Eq. (14) with a work function $\phi=2$eV. Since the emitting area is a flat square, $E_{a}(t)$ refers to the field on the emitter surface. The vacuum field $E_{L}$ is thus $2$V/nm. When emission starts, it is assumed that the field on the emitter is independent of the location on the patch resulting in uniform emission (in reality, the wings have larger current density especially when $L$ is small). Thus $I_{\text{in}}=L^{2}J$ where $J$ is computed using Eq. (14).

When the patch length is large ($L>>D$), the results are expected to mimic the 1-D PIC result reported in Ref. [feng2006, ] where the space-charge affected cathode field is around $1.72$V/nm. The MD simulationtorfason2015 indeed approaches this value when $L=2500$nm. For the model presented here, the anode-area correction is expected to be small for $L=2500$nm and the results should not be very sensitive on the value of $\alpha$. This is indeed found to true with very little variation as $\alpha$ varies from nearly 0 to 1. The smaller $L$ values are however poorly reproduced for standard values of $\alpha$ in the range $[1/4,\sqrt{2}/\pi]$ with $\sqrt{2}/\pi$ being the value appropriate for a square of side-length $L$. Fig. (3) shows a plot of the cathode field variation for $\alpha=1$ for values of $L=2500,1000,900,\ldots,100,50$nm (bottom to top curves). While these do not match perfectly with the MD simulation results (see Fig. 4 of Ref. [torfason2015, ]) especially for $L<500$nm, the trend is very nearly the same and the error is reasonably small.

Planar field emitters require high macroscopic fields in the $E_{0}=3-10$ V/nm range due to the lack of field enhancement ($\gamma_{a}=1$). In practice, field emission occurs from specially designed curved emitters or nano-protrusions on a smooth surface since they require a much smaller macroscopic field. These are thus the natural geometries that need to be investigated for the effect of space charge on field emission.

We shall consider here an axially symmetric curved emitter for which the projected emission area can be considered to be circular of radius $R_{a}$ and the free parameter $\alpha$ can be chosen to be equal to $1/4$. The hemi-ellipsoid is an example of such a geometry. While, the Laplacian problem for the hemi-ellipsoid placed on a conducting plane with the anode far away can be solved analytically smythe ; kosmahl , the presence of space charge makes the problem non-trivial and requires numerical investigation. In order to test our model, we shall consider various magnifications of the basic hemi-ellipsoidal emitter having a height $h=2.515\mu$m and base radius $b=1.5\mu$m. The apex radius of curvature $R_{a}=b^{2}/h\simeq 0.8946\mu$m. The spacing between the anode and cathode plates is $D=10\mu$m. The work function is considered to be $\phi=4.5$eV.

The predictions of the model for $V_{g}=15,17.5$ and $20$kV corresponding to $E_{0}=1.5,1.75$ and $2$V/nm are shown in Fig. 4. While space charge affects the apex field $E_{a}$ at all three values of the macroscopic field, it is strong enough to causes oscillations at $E_{0}=1.75$ and $2$V/nm. The nature of oscillations is similar to the planar case with the period linked to the transit time as observed in planar molecular dynamics simulationstorfason2015 and theoretical modelsjensen2015 . Planar PIC simulations also display oscillationsfeng2006 but these have not been observed yet in 3-D simulations using curved emitters edelen2020 which are considerably more resource intensive.

We shall henceforth compare the model predictions with PIC simulations performed using PASUPAT db_scl ; db_multiscale ; rk2021 with a field emission module based on the cosine law db_parabolic ; db_multiscale ; rk2021 . Our focus will be on the steady state values of the apex field and emitted current and these will be compared with the predictions of the model for various geometric diode parameters. In the PIC simulation using PASUPAT, the hemiellipsoid and cathode plate are considered to be grounded perfect electrical conductors (PECs) while the anode is a PEC at a voltage $V_{g}$. The centre of the hemiellipsoid is at $(X,Y)=(0,0)$ while the transverse boundaries are located at $X,Y=\pm 5\mu$m and have Neumann boundary condition imposed on them. Since the height of the emitter is small compared to the extent of the boundary in the $X$ and $Y$ directions, it is close to being an isolated emitter. The value of $\gamma_{a}$ as calculated using PASUPAT is 4.715 which coincides with the value evaluated using COMSOL. The apex field enhancement effect for an isolated emitter is around $4.8$. Thus, there is a mild shielding effect due to the computational boundary not being very far away.

For the diode described above with $D=10\mu$m, the PIC results for $E_{0}=1.5,1.75$ and $2$V/nm are shown in Fig. 5. The straight line in each case marks the prediction of the model presented in section II for the steady-state apex field. The corresponding result for the emission current is shown in Fig. 6. The agreement is reasonably good especially at the lower values of the macroscopic field where the agreement with the cosine law is goodrk2021 . Note that in order to keep the fluctuations small, the number of time steps per transit time ($\approx 3000)$ has been kept identical for all the simulations.

We shall next consider the effect of transit time by scaling the size of the diode while keeping the field at the apex $E_{a}(0)$ as invariant. We shall first consider a scaling down of all geometric quantities by a factor of 10. Thus, $h=2.515\times 10^{-1}\mu$m, $b=1.5\times 10^{-1}\mu$m $D=10\times 10^{-1}\mu$m and the transverse computational boundaries are at $X,Y=\pm 5\times 10^{-1}\mu$m. In terms of $E_{0}$, the transit time $T_{\text{tr}}\sim(D/E_{0})^{1/2}$. Since the enhancement factor $\gamma_{a}$ is unchanged by the scaling, $E_{a}(0)$ remains the same if $E_{0}$ is maintained at the previous values. Thus $T_{\text{tr}}\sim\sqrt{D}$. Thus, on scaling down the diode by a factor of 10, the transit time decreases by a factor of $\sqrt{10}$ resulting in faster loss of charges from the diode. The effect of space charge is thus expected to be weaker. The PIC simulation results for the scaled down diode are shown in Fig. 7 and Fig. 8. A comparison with Fig. 5 shows the drop in field from the vacuum values to be much smaller indicating a larger value of $\vartheta$. Note that the agreement with the cosine law in the presence of space charge is excellent for this scaled-down dioderk2021 .

We next consider a scaled-up diode to check for the consistency of our results. The scaling factor is 10 so that $h=2.515\times 10\mu$m, $b=1.5\times 10\mu$m $D=10\times 10\mu$m and the transverse computational boundaries are at $X,Y=\pm 5\times 10\mu$m. The results for the time variation of the apex field and injected current are shown in Figs. 9 and 10. The effect of space charge is much stronger as compared to the unscaled case ($D=10\mu$m) thus establishing the importance of transit time in determining the severity of space charge effect on field emission. Not surprisingly, even at $E_{0}=1.5$V/nm, the agreement between the model and the PIC result is not perfect. This coincides with the larger deviation from the cosine law for the scaled-up dioderk2021 .

It is clear from these results that the model is able to predict the steady-state space-charge affected apex field $E_{P}$ and the injected current $I_{\text{in}}$ reasonably well, when the space-charge effect is moderate. We can use the values obtained from the model to quantify the severity of space-charge. Fig. 11 shows a plot of $\vartheta$ as a function of $E_{L}$ for the model considered where $D=10\mu$m and the apex enhancement factor is $\gamma_{a}=4.715$. Note that the upper limit of the vacuum field used in the Fig. 11 and 13 is $E_{L}\simeq 13.2$V/nm. This is to ensure that the top of the potential barrier remains above the Fermi level. The values of $\vartheta$ for $E_{L}$ exceeding 10V/nm or $\vartheta<0.8$ are not very accurate due to larger deviations from the cosine law. They are however expected to be indicative of the general trend.

The model results can also be cast in a (${\xi},\tilde{\theta}$) plot for completeness. This is shown in Fig. 12 for $D=10\mu$m. Also shown is the straight line of Eq. (10) with the parameter $\tilde{\omega}\approx 1.33$, obtained by a least square fit.

Finally, the data obtained using the space-charge affected field emission model is shown as an FN plot in Fig. 13. Clearly, the space charge affected field emission current deviates from the straight line as observed earlier barbour53 ; jensen97a ; jensen97b ; forbes2008 . Also shown alongside are FN plots of 2 different space charge limited currents. The first of these (solid triangles in Fig. 13) is the planar Child-Langmuir current from an area equal to the base of the curved emitter. Thus, $I=J_{\text{CL}}\pi b^{2}$ where

The space-charge affected field emission current can thus exceed the planar space charge limit but is bounded by the space charge limited current for a curved emitter.

The authors have no conflicts to disclose.

The data that supports the findings of this study are available within the article.

Data Availability: The data that supports the findings of this study are available within the article.