Universal Velocity-Field Characteristics for a nanowire of arbitrary degeneracy

A nanowire (NW), as shown in Fig. 1, is an example of quasi-one-dimensional nanostructure where two of the three Cartesian directions are quantum confined with standing waves. The third direction is quasi-free with propagating carrier wave with charge q as an information carrier [1]. The presence of a high electric field in the quasi-free direction orients the otherwise stochastic velocity vectors in equilibrium to those streamlined unidirectional one in a very high electric field, making the nonequilibrium distribution highly asymmetric. In a mean free path l, the energy qE→·l→ on a tilted band diagram in an electric field E→ modifies the Fermi-Dirac distribution to yield [2] ƒ (E) =1/[1+exp((E−E F + qE→·l→)/ k B T)] Here E is the energy of a drifting carrier, E F the Fermi energy, and T is the lattice temperature. In equilibrium the number of carriers going in either (+ve or −ve) direction along the quasi-free direction are equal because of their stochastic motion. However, this equilibrium is disturbed as the number n + of carriers drifting in the +x-direction increase at the cost of n− going in the -x-direction, as shown in Fig. 2. The reduced Fermi energy η = (E F − E C )/k B T is now a function of normalized electric field, where E co = V t /l o is the critical electric field with V t =k B T/q is the thermal voltage with value 25.9 mV at room temperature. η also depends on the degeneracy level C 1 = n 1 /N C1 where n 1 is the linear carrier concentration per unit length and N C1 is the effective density of states in the nanowire. Fig. 3 shows η as a function of δ o for 4 degeneracy levels C 1 =0.1, 1, 2, and 3. C 1 =0.1 follows the nondegenerate statistics as expected for a low-carrier concentration. η for a strongly degenerate concentration first rises, as more quantum states in the +ve direction populate in concert with Pauli Exclusion Principle and then decreases linearly with δ o . In the extreme limit of degeneracy, only N C1 /2 states are available for filling electrons in the positive direction. The Fermi level rises as quantum states N C1 /2 accommodate additional electrons in the +ve direction. The excess fraction of electrons Δn 1+ / n 1 = (n 1+ −n 1− )/n 1 moving in the +ve direction is shown in Fig. 4 for the 4 degeneracy levels. In an extremely high electric field, almost all electrons are moving in the +ve direction, giving the net drift that saturates to the intrinsic velocity appropriate for effective number of quantum states N C1 /2.