Electric currents at semiconductor surfaces from the perspective of drift-diffusion equations

Surface sensitive electric current measurements are important experimental tools poorly corroborated by theoretical models. We show that the drift-diffusion equations offer a framework for a consistent description of such experiments. The current flow is calculated as a perturbation of an equilibrium solution depicting the space charge layer. We investigate the accumulation and inversion layers in great detail. Relying on numerical findings, we identify the proper length parameter, the relationship of which with the length of the space charge layer is not simple. If the length parameter is large enough, long-ranged modes dominate the Green's function of the current equation, leading to two-dimensional currents. In addition, we demonstrate that the surface behavior of the currents is ruled by only a few parameters. This explains the fact that simplistic conductivity models have proven effective but makes reconstructions of conductance profiles from surface currents rather questionable.

Figure 1

Panel (a) shows the three conductivity profiles numerically considered: $14(1-tanh\frac{z-0.06}{0.005})-9(1-tanh\frac{z-0.01}{0.003})+1$—red, $25(1-tanh\frac{z-0.0405}{0.015})-21(1-tanh\frac{z-0.01}{0.01})+1$—blue, and $15(1-tanh\frac{z-0.048}{0.007})+1$—green. The resulting potentials $U\left(z\right)$ are shown in panel (b); the following functions ${\psi }^2(k;0)$ in panel (c). Image (d) is a close-up view of ${\psi }^2(k;0)$ for small $k$ with several points corresponding to the Lorentzian, as defined in Eq. (32), with ${\mathrm{\Gamma }}^2=0.41\mu {\text{m}}^{-2}$, $p=0$, and $A=1$ shown for illustration. The functions ${\psi }^2(k;0)$ are rescaled so that ${\psi }^2(0;0)=1$ [panels (c) and (d)].

Figure 2

Normalized resistance $\omega =\sigma \left(\infty \right)\varphi /I$ for the geometry depicted in Sec. 3 for various ${\mathrm{\Gamma }}^2$: ${\mathrm{\Gamma }}^2=0.0005\mu {\text{m}}^{-2}$ in (a), ${\mathrm{\Gamma }}^2=0.1\mu {\text{m}}^{-2}$ in (b), and ${\mathrm{\Gamma }}^2=20\mu {\text{m}}^{-2}$ in (c). Colors correspond to different values of $D$: $2\mu \text{m}$—green, $10\mu \text{m}$—blue, and $20\mu \text{m}$—red. The two-dimensional character resulting in the independence of $D$ is shown in panel (a). The nearly three-dimensional case is shown in panel (c) as the quantity $DR\sigma \left(\infty \right)=D\omega$ appears to be $D$ independent. An exemplary transition behavior is shown in panel (b), where neither graphs of $\omega \left(x\right)$ nor $D\omega \left(x\right)$ coincide. The points (stars) in panels (a) and (c) demonstrate the validity of Eq. (21) and Eq. (22), respectively, with fitted numerical coefficients. $D\omega$ is a dimensionless quantity while $\omega$ has the dimension ${\text{length}}^{-1}$, we write ${\text{cm}}^{-1}$ due to the convention $\left[\sigma \right]={\left(\mathrm{\Omega }\text{cm}\right)}^{-1}$.

Figure 3

Upper panel: Logarithmic variation of the equilibrium electron and hole densities normalized by the bulk carrier density $\tilde{n}={log}_{10}\frac{n_0\left(z\right)}{p_b+n_b}$ and $\tilde{p}={log}_{10}\frac{p_0\left(z\right)}{p_b+n_b}$ at 300 K and 120 K. The bulk carrier density reads $6\times {10}^{15}$ and $5\times {10}^{15}$ ${\mathrm{cm}}^{-3}$ at 300 and 120 K, respectively. Lower panel: The number of holes located between the surface and the depth $z$ as a fraction of all the positive current carriers in the sample. The exact value of the sample thickness $d$ in the range 100–1000 $\mu \mathrm{m}$ has no impact on the graphs. These graphs demonstrate that half of the holes are located in the 5 nm zone beneath the surface, while nearly all positive charge carriers are located in the 20 nm subsurface zone at room temperature. At 120 K the holes are concentrated even closer to the surface.