Abstract
For charge injection from an electrode into a trap-filled dielectric slab, its current–voltage ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${I}$ </tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${V}$ </tex-math></inline-formula> ) characteristics are governed by the Mark–Helfrich (MH) law. By matching the experimentally measured <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${I}$ </tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${V}$ </tex-math></inline-formula> characteristics to a right <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${I}$ </tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${V}$ </tex-math></inline-formula> model, one can characterize the microscopic properties of the dielectric like its carrier mobility and traps distribution. The original MH law was developed for a bulk solid and may not be valid for modern ultrathin dielectrics used in 2-D electronics. Here, we revise the MH law for an ultrathin trap-filled dielectric of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L}$ </tex-math></inline-formula> biased with a voltage of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${V}$ </tex-math></inline-formula> . Our model suggests a new scaling of the current line density: <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {J}_{\text {2-D}} \,\,\propto \,\,{[}{({V}}/{\alpha {L}{)}} \text {exp}{(} - {(\beta {l}}/{l+{1}}{)}{)} {]}^{l+{1}}$ </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">$\alpha $ </tex-math></inline-formula> = (2.8, 2.03) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> = (1.02, 0.94) are numerical values for two different geometrical (edge, strip) contacts, respectively. Using this 2-D ultrathin MH law, we demonstrate that the estimated carrier mobility can be significantly different from the traditional MH law. Under the same material properties, our model also highlights that strip contact geometry will always lead to a larger current flow than edge contact geometry. Thus, the developed model should be useful for the characterization of the ultrathin dielectrics used in 2-D materials-based electronics, organic semiconductors, and thin-film electronics.
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