The Implications of Collisions on the Spatial Profile of Electric Potential and the Space-Charge-Limited Current

The space-charge-limited current (SCLC) in a vacuum diode is given by the Child-Langmuir law (CLL), whose electric potential <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\phi }\text {(}{x}\text {)}\propto \text {(}{x} /{D}\text {)}^{{4} /{3}}$ </tex-math></inline-formula>, where x is the spatial coordinate across the gap and D is the gap separation distance. For a collisional diode, SCLC is given by the Mott-Gurney law (MGL) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\phi }\text {(} {x}\text {)}\propto \text {(}{x} /{D}\text {)}^{{3} / {2}}$ </tex-math></inline-formula>. Here, we apply a capacitance argument for SCLC and use the transit time from a recent exact solution for collisional SCLC to show that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\phi }\text {(}{x}\text {)}\propto \text {(}{x}/{D}\text {)}^{\xi }$ </tex-math></inline-formula> for a general collisional gap, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${4} / {3}\le \xi \le {3}/ {2}$ </tex-math></inline-formula>. Furthermore, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\xi $ </tex-math></inline-formula> is strictly a function of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\nu {T}$ </tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\nu $ </tex-math></inline-formula> is the collision frequency and T is the electron transit time. Using this definition of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\xi $ </tex-math></inline-formula>, we estimate the spatial dependence of the electron velocity and use the gap capacitance to derive an analytic equation for collisional SCLC that agrees within ~4.5% of the exact solution that requires solving parametrically through T. This analytic equation for general <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\xi $ </tex-math></inline-formula> asymptotically recovers the CLL as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\nu }\to {0}$ </tex-math></inline-formula> and the MGL as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\nu } \to \infty $ </tex-math></inline-formula>. Matching these limits shows that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\xi }\approx {1.40}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${V}\propto {D}^{{2}}{\nu }^{{2}}$ </tex-math></inline-formula> at the transition from a vacuum to a collisional diode for any device condition.