Electronic transport through Si nanowires: Role of bulk and surface disorder

We calculate the resistance and mean free path in long metallic and semiconducting silicon nanowires (SiNW’s) using two different numerical approaches: a real-space Kubo method and a recursive Green’s-function method. We compare the two approaches and find that they are complementary: depending on the situation a preferable method can be identified. Several numerical results are presented to illustrate the relative merits of the two methods. Our calculations of relaxed atomic structures and their conductance properties are based on density functional theory without introducing adjustable parameters. Two specific models of disorder are considered: Unpassivated, surface reconstructed SiNW’s are perturbed by random on-site (Anderson) disorder whereas defects in hydrogen passivated wires are introduced by randomly removed H atoms. The unpassivated wires are very sensitive to disorder in the surface whereas bulk disorder has almost no influence. For the passivated wires, the scattering by the hydrogen vacancies is strongly energy dependent and for relatively long SiNW’s $(L>200\mathrm{nm})$ the resistance changes from the Ohmic to the localization regime within a $0.1\text{−}\mathrm{eV}$ shift of the Fermi energy. This high sensitivity might be used for sensor applications.

Figure 1

Top: schematic time evolution of a random phase state $\mid {\psi }_r\left(t\right)⟩$ initially located in the central region of the wire. Middle: the geometry in the Kubo method consists only of a large device region. Bottom: in the GF approach a device region is connected to two semi-infinite leads.

Figure 2

The device region $\left(D\right)$ is divided into $M$ subcells. The subcells are so large that they only couple to the nearest-neighbor cells. Also, the left $\left(L\right)$ and right $\left(R\right)$ leads consisting of equal unit cells, described by ${\mathbf{H}}_0$, couple only to the first and last cell of the device region, respectively.

Figure 3

(Color online) Top: schematic illustration of supercells containing five unit cells from three different DFT calculations: one pristine wire and two wires with different defects. In the latter the defects are located in unit cell number 3. The cells number 1 and 5 are assumed to be the same in all three calculations. Bottom: a long wire is constructed by joining pieces from the different DFT calculations.

Figure 4

(Color online) (a): Side view of the unpassivated SiNW containing five unit cells, separated by the red dashed lines. (b): End view of the unpassivated SiNW. (c): Band structure around the Fermi energy (dashed line). Four bands are crossing the Fermi energy, two being degenerate (with $E-E_F=-0.05\mathrm{eV}$ at the $\Gamma$ point).

Figure 5

(Color online) Time-dependent diffusion coefficient at $E-E_F=0.1\mathrm{eV}$. The bulk disorder has little effect and the transport is ballistic. The edge-disordered system shows diffusive behavior with a constant diffusion coefficient for $t>150\mathrm{fs}$.

Figure 6

(Color online). (a): mean free path $l_e$ vs energy $E-E_F$ for $\Delta \epsilon =0.4\mathrm{eV}$. The solid line is obtained with the Kubo method, and the GF results are marked by squares while the dashed line shows the FGR results. GF results are average values for 200 different samples while the Kubo results are mean values of 10 different samples. (b): scaling of $l_e$ vs $1∕\Delta \epsilon$ shown on a logarithmic scale. Circles, squares, and triangles are calculated with GF while the lines are obtained using FGR at the same energies.

Figure 7

(Color online) (a): cross section of the H-passivated wire. (a): energy-dependent conductance of a pristine wire and of wires containing a single vacancy of number 1–3.

Figure 8

(Color online) Mean free path vs energy for a concentration of vacancies corresponding to $⟨d_H⟩=5.6\mathrm{nm}$. The solid black line corresponds to wires containing all possible vacancies. The dash-dotted blue line is obtained using Matthiessen’s rule for single-scattering events and the dashed red curve corresponds to a wire with Anderson disorder $(\Delta \epsilon =1.3\mathrm{eV})$.

Figure 9

(Color online) Length-dependent resistance at the energies $E=-0.3\mathrm{eV}$ (dashed blue line), $E=-0.15\mathrm{eV}$ (dash-dotted green line), and $E=-0.03\mathrm{eV}$ (solid red line). The average distance between vacancies is $⟨d_H⟩=2.8\mathrm{nm}$. The inset shows the scaling of $l_e$ vs average distance $⟨d_H⟩$ between vacancies at the same three energies as above.

Figure 10

(Color online) (a): model system. We consider only nearest-neighbor interaction with the tight-binding parameters ${\gamma }_1$ and ${\gamma }_2$ corresponding to hopping in the chain direction and in perpendicular direction, respectively. The on-site energy is ${\epsilon }_0$. (b): band structure for the two bands separately. Notice the definition of the energies $E_1-E_4$.

Figure 11

(Color online). (a): the numerically calculated conductance (dash-dotted black line) is plotted together with the two analytical results $g^{\left(1\right)}\left(E\right)$, Eq. (A6) (dashed blue line) and $g^{\left(2\right)}\left(E\right)$, Eq. (A7) (solid red line). (b): zoom-in around $E=E_3$, showing that $g^{\left(1\right)}\left(E\right)$ closely resembles the numerical result.