Ultrashort-Pulse Child-Langmuir Law in the Quantum and Relativistic Regimes

This Letter presents a consistent quantum and relativistic model of short-pulse Child-Langmuir (CL) law, of which the pulse length $\tau$ is less than the electron transit time in a gap of spacing $D$ and voltage $V$. The classical value of the short-pulse CL law is enhanced by a large factor due to quantum effects when the pulse length and the size of the beam are, respectively, in femtosecond duration and nanometer scale. At high voltage larger than the electron rest mass, relativistic effects will suppress the enhancement of short-pulse CL law, which is confirmed by particle-in-cell simulation. When the pulse length is much shorter than the gap transit time, the current density is proportional to $V$, and to the inverse power of $D$ and $\tau$.

Figure 1

The normalized short-pulse quantum CL law in terms of the classical CL law (${\Gamma }_Q$), and the pulse width of the electron sheet ($\zeta$ [nm]), as a function of $X_{\mathrm{CL}}=\tau /T_{\mathrm{CL}}$ in a large gap for (a) $D=1\mathrm{cm}$ and $V_g=1\mathrm{kV}$, and (b) $D=1\mathrm{mm}$ at $V_g=1$ and 10 kV. The dashed lines in (a) are the short-pulse classical CL law.

Figure 2

The normalized short-pulse quantum CL law in terms of the classical CL law (${\Gamma }_Q$) as a function of $X_{\mathrm{CL}}=\tau /T_{\mathrm{QCL}}$ in a nano gap for (a) $D=10\mathrm{nm}$ at $V_g=1$ to 100 V (top to bottom), and (b) $V_g=10\mathrm{V}$ at $D=1$ to 100 nm (top to bottom). The dashed lines are the short-pulse classical CL law.

Figure 3

The normalized short-pulse relativistic CL law in terms of the classical CL law (${\Gamma }_R$) as a function of (a) $X_{\mathrm{CL}}=\tau /T_{\mathrm{RCL}}$ for $U=1$ to 100 (top to bottom), and (b) $U$ for $X_{\mathrm{CL}}=0.01$ to 1 (top to bottom). The symbols in (b) are the PIC simulation results. (c) The enhancement of the short-pulse relativistic SCL current density over the long-pulse limit $J_{\mathrm{RCL}}[X_{\mathrm{CL}}]/J_{\mathrm{RCL}}$ [1] as a function of $X_{\mathrm{CL}}$ for $U=1$ to 100 (top to bottom). The dashed lines are the short-pulse classical CL law.