New Scaling of Child-Langmuir Law in the Quantum Regime

This paper presents a consistent quantum mechanical model of Child-Langmuir (CL) law, including electron exchange-correlation interaction, electrode’s surface curvature, and finite emitter area. The classical value of the CL law is increased by a larger factor due to the electron tunneling through the space-charge potential, and the electron exchange-correlation interaction becomes important when the applied gap voltage $V_g$ and the gap spacing $D$ are, respectively, on the order of Hartree energy level, and nanometer scale. It is found that the classical scaling of $V_g^{3/2}$ and $D^{-2}$ is no longer valid in the quantum regime, and a new scaling of $V_g^{1/2}$ and $D^{-4}$ is established. The smooth transition from the classical regime to the quantum regime is also demonstrated.

Figure 1

The normalized quantum CL law ${\mu }_Q$ as a function of $\lambda$ for various ${\varphi }_g={10}^{-2}$ to ${10}^2$ (solid lines: top to bottom), ${\varphi }_g\gg 1$ (dashed line), and classical limit (dash-dotted line).

Figure 2

The values of ${\mu }_Q{V}_g^{3/2}$ as a function of gap voltage $V_g$ (volt) for various gap spacing $D=1$, 10, and 100 nm (solid lines: top to bottom). The dashed lines are without an exchange-correlation term (${\varphi }_{\mathrm{x}\mathrm{c}}=0$) for $D=10$ and 100 nm (top to bottom), and the short-dashed line is the classical limit.

Figure 3

(a) The normalized quantum CL law ${\mu }_Q$ as a function of gap spacing $D$ (nm) for various values of $ϵ=(E-E_F)/e{V}_g=0.5$ to $-0.5$ at gap voltage $V_g=1\mathrm{V}$. (b) The boundary of the transition from the quantum region (small $D$) to the classical region (large $D$) for$E-E_F[\mathrm{e}\mathrm{V}]=-0.2$, $-0.3$, and $-0.5$.

Figure 4

(a) The ratio of ${\mu }_Q[N]$ to ${\mu }_Q[N=0]$ as a function of the normalized electrode’s curvature $R$ at gap voltage $V_g=1\mathrm{V}$ for gap spacing: $D=10\mathrm{n}\mathrm{m}$ (solid line) and $D=100\mathrm{n}\mathrm{m}$ (dashed line), where $N=1$ and $N=2$ are cylindrical and spherical geometry, respectively. The symbols are for $V_g=10\mathrm{V}$ and $D=10\mathrm{n}\mathrm{m}$: triangle ($N=2$) and rectangle ($N=1$). (b) The mean location of electrons $\rho$ as a function of gap voltage $V_g$ (volt) for various values of gap spacing $D=10$, 100, and 1000 nm. The classical limit is $\rho =0.25$ (dashed line).