Monte Carlo calculations for metal-semiconductor hot-electron injection via tunnel-junction emission

We present a detailed description of a scheme to calculate the injection current for metal-semiconductor systems using tunnel-junction electron emission. We employ a Monte Carlo framework for integrating over initial free-electron states in a metallic emitter and use interfacial scattering at the metal-semiconductor interface as an independent parameter. These results have implications for modeling metal-base transistors and ballistic electron emission microscopy and spectroscopy.

Figure 1

Schematic diagram of the ballistic electron-injection process. The “X” shows the location of an example $k$-space state, which conserves both parallel momentum and energy throughout the process.

Figure 2

Schematic tunnel-junction energy diagram.

Figure 3

Energy distribution of tunneling electrons from Au through a vacuum tunnel barrier with $1\mathrm{V}$ across it. The nonzero temperature $\left(300\mathrm{K}\right)$ causes a smearing of the distribution near the cusp corresponding to the emitter Fermi energy.

Figure 4

Monte Carlo calculation of tunnel-junction current-voltage spectrum. The barrier width is $6\mathrm{\AA }$ and the height is $5\mathrm{eV}$ above the Fermi energy $\left(5\mathrm{eV}\right)$.

Figure 5

Schematic Interface Brillouin Zone for GaAs(100). The arrow on the upper right indicates parallel momentum compatible with available semiconductor states, while the arrow in the lower left indicates parallel momentum at which there are no available states.

Figure 6

A 2D projection of the shelllike integration region used in the calculation of the tunnel junction current.

Figure 7

Integral calculation of tunnel junction current-voltage spectrum. The barrier width is $6\mathrm{\AA }$ and the height is $5\mathrm{eV}$ above the Fermi energy $\left(5\mathrm{eV}\right)$. Also shown for comparison are the results of the previously presented Monte Carlo method.

Figure 8

Monte Carlo BEES calculation for $\mathrm{Au}∕\mathrm{Ga}\mathrm{As}$ at $300\mathrm{K}$. This simulation uses ${10}^8$ samples of $k$ space.

Figure 9

A schematic of the “Fermi sea” in the tip metal. The arrow on the left represents the Fermi wave vector $k_F$ and the arrow on the right represents the minimum electron wave vector magnitude [Eq. (33)]. At a tip bias above the Schottky barrier, only electrons within a shell (dark shaded region) between spheres of the two radii have enough energy to couple with states in the semiconductor.

Figure 10

(a) The tunnel process gives rise to a momentum distribution with a characteristic width (see Ref. 19, Fig. 3). (b) This distribution is compared to the projection of the available semiconductor states.

Figure 11

Scattering probability (SP) changes the relative contributions of the $\Gamma$ and $L$ valleys in GaAs(100) to reflect experimental observations.