Ballistic structure in the electron distribution function of small semiconducting structures: General features and specific trends

In the hot-electron, ballistic transport regime, we have calculated the distribution function f(v,x) for electrons in a submicron $N^{+}$-$N^{\mathrm{-}}$-$N^{+}$ GaAs structure by directly solving the coupled Poisson and Boltzmann equations using simple relaxation-time models. Ballistic electrons cause both the dominant peak in f(v,x) throughout much of the $N^{\mathrm{-}}$ region and additional structure, ballistic echoes, because of intervalley transfer. Both phenomena can be traced to simple features of the potential energy curve which should occur in many device geometries. The structure in the distribution function responds to changes in the parameters—mobility, voltage, and lattice temperature—in the way expected for ballistic electrons: a larger mobility, a larger voltage, or a lower temperature makes the structure more prominent. By integrating over the distribution function, we calculate as a function of distance the current and the width of the distribution, expressed as an effective ‘‘temperature.’’ The I-V characteristics are nearly linear for all the cases studied (as well as for the experiment) showing that the current is not a sensitive probe of ballistic structure. In fact, the current as a function of mobility saturates in the ballistic regime (high mobility) where the injection over the initial barrier determines the current. The distribution functions we calculate are far from being drifted Maxwellian in form so a thermodynamic temperature cannot be defined. However, in the near-equilibrium regime we show that ‘‘heating’’ and ‘‘cooling’’ are first order in the applied voltage in contrast to bulk behavior where heating is second order in the field. At higher voltages, the width of the distribution function varies dramatically from less than the equilibrium width at injection to much larger than the equilibrium width.