Effect of growth orientation and surface roughness on electron transport in silicon nanowires

We report a study of the effect of growth orientation and surface roughness on electron transport in small-diameter hydrogen passivated silicon nanowires (NWs). We employ a nonequilibrium Green’s function technique within an $s{p}^3{d}^5{s}^{*}$ tight-binding approximation to show that band structure strongly affects current-voltage characteristics of ideal NWs, leading to current falloff at high drain bias for certain growth orientations. Surface roughness suppresses small bias conductance and current, and leads to a nonmonotonic dependence on gate bias. We also find that surface roughness results in a decrease of current with the length of a NW. The rate of the decrease depends on the growth directions and diameter. As the diameter of a NW becomes smaller, a transition of electron transport to Anderson localization regime may occur even at room temperature.

Figure 1

(Color online) Cross sections of silicon nanowires grown in [110], [111], and [100] directions passivated with hydrogen atoms. The diameters are 1.26, 1.52, and $1.56\mathrm{nm}$, respectively.

Figure 2

(Color online) $s{p}^3{d}^5{s}^{*}$ tight-binding band structure of NWs shown in Fig. 1. The dashed line shows a position of the Fermi level in the wire at zero bias. The upper row corresponds to $n$-type and the lower row to the $p$-type wires. The widths of conducting subbands $c_1$ ($n$-type) and $v_1$ ($p$-type) determine the maximum bias under which the wire would conduct.

Figure 3

(Color online) The (a) $s{p}^3{d}^5{s}^{*}$ tight-binding and (b) DFT-GGA band structure for the [111] NW. The splitting of subbands in $n$-type [111] NW at $ka=\pi$ is zoomed in.

Figure 4

(Color online) Transmission versus energy in ideal wires under applied drain bias $V_d$. Linear voltage drop is assumed. The arrows show the position of the Fermi energy $E_{Fs}$ in the source contact, while the Fermi energy in the drain is $E_{Fd}=E_{Fs}-V_d$. (a) $T\left(E\right)$ in the transport energy window $[E_{Fd};E_{Fs}]$ is not changed by applied bias in $n$-type [110] NW. (b) In $n$-type [111] wire, $T\left(E\right)$ decreases at a bias higher that $350\mathrm{mV}$. (c) In $n$-type [100] wire, transmission decreases dramatically when the bias exceeds $800\mathrm{mV}$.

Figure 5

(Color online) Drain current-drain voltage characteristics of ideal NWs shown in Fig. 1. The current falls off whenever bias exceeds the width of the conducting subband in Fig. 2. (a) both $n$- and $p$-type [110] NWs show perfect saturation of drain current, (b) In $n$-type [111] wire, the current shows a dramatic falloff when the drain bias exceeds $350\mathrm{mV}$. (c) In $n$- and $p$-type [100] wires, the current falls off when the bias exceeds $800\mathrm{mV}$. The current does not vanish completely due to intersubband coupling.

Figure 6

(Color online) Saturation current at a drain bias of $1\mathrm{V}$ versus length of ideal [110] (solid) and [111] (dashed) NWs shown in Fig. 1. Energy splitting in the band structure of [111] nanowire causes a decrease of current in long nanowires. The onset of intersubband tunneling helps conduction and causes a recovery of current in short wires.

Figure 7

(Color online) A cross section and a side view of $1.26\text{−}\mathrm{nm}$-thick [110] silicon nanowire with introduced “white noise” surface roughness and passivated with hydrogen. One atomic layer is added with probability $1∕2$ in each unit cell at each facet of the wire.

Figure 8

(Color online) Logarithm of the density of states in (a) conduction and (b) valence bands as a function of energy and coordinate along the axis of the $1.26\text{−}\mathrm{nm}$-thick [110] wire with surface roughness. Density of states was summed over all atoms lying in a plane perpendicular to the axis. The spatial inhomogeneity of DOS causes reflection of electrons and decreases the current.

Figure 9

(Color online) Zero bias conductance $G\left(E_F\right)$ versus Fermi energy in $\sim 100\mathrm{nm}$ rough NWs, computed using Eq. (3). Conductance in rough NWs takes noninteger values and decreases in value. The decrease is more significant in the valence band, where the number of subbands increases more rapidly with energy. The effect of roughness is much stronger at high electron or hole energies due to a larger number of subbands and thus stronger intersubband mixing. Conductance shows local minima at energies corresponding to subband bottoms.

Figure 10

(Color online) Saturation current at a drain bias of $1\mathrm{V}$ as a function of wire length for different growth directions. (a) The decrease of current in the [110] nanowire with length is more significant for the $p$-type case. (b) Current in the [111] wire shows similar trend to the [110] case, but the decrease of current is more significant. (c) Current in the [100] wire shows a dramatic decrease with length both for $n$- and $p$-type cases. One can see that $100\mathrm{nm}$ $n$- and $p$-type [100] NWs lose 85% of current.

Figure 11

(Color online) Saturation current of $n$-type [110] wires of 1.26 and $0.46\mathrm{nm}$ diameter at a bias of $1\mathrm{V}$. The number of subbands in the transport energy window has decreased from 20 in the $1.26\mathrm{nm}$ wire to 4 in the $0.46\mathrm{nm}$ wire. When the wire length exceeds $100\mathrm{nm}$, the current decreases exponentially with length as a manifestation of Anderson localization. Fitting with exponential function (dotted lines) gives an estimate of localization lengths of $380\mathrm{nm}$ for the $1.26\mathrm{nm}$ wire and $12\mathrm{nm}$ for the $0.46\mathrm{nm}$ wire.

Figure 12

(Color online) Saturation current of $0.46\mathrm{nm}$ [110] NW at $1\mathrm{V}$ in the presence of electron-phonon scattering (solid line) compared to the case of no scattering (dashed line). For wires shorter than the phase-breaking length $L_{\varphi }\sim 40\mathrm{nm}$, electron-phonon scattering results in a slight decrease of current. In longer wires, the electron-phonon scattering causes an increase of current and recovering from Anderson localization regime. Inset: Deformation potential of [110] ideal NW as a function of diameter. The electron-phonon coupling strength in small-diameter nanowire is much smaller than in bulk Si. In the $0.46\mathrm{nm}$ wire, with Fermi velocity $v_F\sim 5\times {10}^5\mathrm{m}∕\mathrm{s}$, the mean free path ${\lambda }_{ph}$ is estimated to be around $5\mu \mathrm{m}$.