Appearance of effective surface conductivity: An experimental and analytic study

Surface conductance measurements on $p$-type doped germanium show a small but systematic change to the surface conductivity at different length scales. This effect is independent of the structure of the surface states. We interpret this phenomenon as a manifestation of conductivity changes beneath the surface. This hypothesis is confirmed by an analysis of the classical current flow equation. We derive an integral formula for calculating the effective surface conductivity as a function of the distance from a point source. Furthermore, we derive asymptotic values of the surface conductivity at small and large distances. The actual surface conductivity can only be sampled close to the current source. At large distances, the conductivity measured on the surface corresponds to the bulk value.

Figure 1

Conductivity $\sigma$ as a function of the distance $D$ between the current source and drain for three different germanium surfaces. The measurements were done using the four-point–probe technique in a collinear arrangement outlined in detail in Ref. [11]. As some points coalesce, the data are also given in Table 1.

Figure 2

A schematic picture explaining the band bending and associated change in the number of electrons and holes. The acronym s.s. stands for surface states, while $n_e$ and $n_h$ denote the electron and hole density, respectively. The width of the subsurface region depends on band bending and bulk doping, which is usually in the range given in the picture. The depicted situation with upward band bending corresponds to germanium, where the valence band appears very close to the Fermi level.

Figure 3

Gaussian conductance profile describing a tiny enhancement of the conductance at the surface (red dotted line) and related potential $V$ (black solid line). The conductance profile reads $\sigma \left(z\right)=\sigma \left(0\right)\left[0.833+0.267exp(-0.0002z^2)\right]$.

Figure 4

Gaussian conductance profile describing a small deficiency of the conductance at the surface (red dotted line) and related potential $V$ (black solid line). The conductance profile reads $\sigma \left(z\right)=\sigma \left(0\right)\left[1.25-0.25exp(-0.0002z^2)\right]$.

Figure 5

The black solid curve corresponds to the function $K_0\left(u\right)$ with a logarithmic divergence close to $u=0$ and vanishing exponentially for $u\to \infty$. Two other curves correspond to some function ${\psi }^2(u/r;0)$ with $r=0.6$ (red and dashed) and $r=12$ (blue dotted). Due to the quick decrease of $K_0\left(u\right)$, only the region $(0;3.5)$ contributes significantly to the integral, Eq. (14). For small values of $r$, the function ${\psi }^2$ gets compressed and it has the limiting value 1 nearly on the whole interval $(0,3.5)$. For large values of $r$, the function ${\psi }^2$ is stretched, so that for sufficiently large $r$ it is nearly constant [equal to $\psi (0;0)]$ on the interval. The compression and stretching are responsible for the obtained asymptotic behavior.

Figure 6

Numerically obtained ${\sigma }_{\text{eff}}\left(r\right)$ for the conductivity $\sigma \left(z\right)$ described in Fig. 3. The dotted horizontal line corresponds to the asymptotic value $\sigma \left(z\right)$ with $z\to \infty$. The inset shows ${\psi }^2(k;0)$.

Figure 7

Numerically obtained ${\sigma }_{\text{eff}}\left(r\right)$ for the conductivity $\sigma \left(z\right)$ described in Fig. 4. The dotted horizontal line corresponds to the asymptotic value of the bulk conductivity. The inset shows ${\psi }^2(k;0)$.

Figure 8

Experimental resistance multiplied by $D$ measured for various $s$ for sample A with a passivated surface. For $s<0.2$, all the data coincide. The data for $D=8$ and $16\mu \mathrm{m}$ coincide for nearly all values of $s$. Solid lines show numerically obtained data for the profile $\sigma \left(z\right)=\left(4.5{cosh}^{-1}\frac{z}{120}+1\right)\sigma \left(\infty \right)$.