2.1 The AVC Method

The AVC method was developed to measure the depth of a pn-junction formed by implantation. Fig. 2.1 shows a part of a cross-sectioned semiconductor device and the simplified geometry in which the doping varies only in one direction, used in the further discussion and for simulations. In a real measurement of course a two-dimensional cross section is used in which the doping variation is not limited to one direction.

Figure 2.1: Part of a cross-sectioned device and the simplified geometry used for simualtion.

During an AVC measurement a beam of high energy primary electrons is focused on the surface of a cross-sectioned semiconductor device. Typical electron beam energies are of the order of several keV and the beam diameter is of the order of several 10 nm. The injected electrons generate a high number of secondary electron-hole pairs by impact ionization. A fraction of the secondary electrons has enough energy to leave the semiconductor (see Fig. 2.2) and their kinetic energy can be measured. From the kinetic energy information regarding the net doping can be extracted. By scanning the electron beam across the semiconductor surface the two-dimensional doping distribution is probed.

Figure 2.2: A beam of high energy primary electrons generates secondary electron-hole pairs by impact ionization.

According to the semi-classical theory a potential applied to a semiconductor causes the energy levels to shift proportional to the potential. In a nonuniformly doped semiconductor the ionized dopants and the connected built-in potential cause such band bending. The built-in potential $\varphi_{\mathrm{bi}}^{}$ across a pn-junction is defined as

$$\varphi_{\mathrm{bi}}^{} {\frac{\ensuremath{\mathrm{k_{B}}}\cdot T}{\mathrm{q}}} \left(\vphantom{\frac{1}{2\cdot N_{\mathrm{C}}\cdot\exp\left(-\fr... ...ath{E_{\mathrm{g}}}}{\ensuremath{\mathrm{k_{B}}}\cdot T}\right)}\right)}\right. {\frac{1}{2\cdot N_{\mathrm{C}}\cdot\exp\left(-\frac{E_{C}}{\ensuremath{\mathrm{k_{B}}}\cdot T}\right)}} \left(\vphantom{N_{N} + \sqrt{N_{N}^{2} + 4 \cdot N_{\mathrm{C}}\... ...ensuremath{E_{\mathrm{g}}}}{\ensuremath{\mathrm{k_{B}}}\cdot T}\right)}}\right. \sqrt{N_{N}^{2} + 4 \cdot N_{\mathrm{C}}\cdot N_{\mathrm{V}}\cdot... ...-\frac{\ensuremath{E_{\mathrm{g}}}}{\ensuremath{\mathrm{k_{B}}}\cdot T}\right)} \left.\vphantom{N_{N} + \sqrt{N_{N}^{2} + 4 \cdot N_{\mathrm{C}}\... ...ensuremath{E_{\mathrm{g}}}}{\ensuremath{\mathrm{k_{B}}}\cdot T}\right)}}\right) \left.\vphantom{\frac{1}{2\cdot N_{\mathrm{C}}\cdot\exp\left(-\fr... ...ath{E_{\mathrm{g}}}}{\ensuremath{\mathrm{k_{B}}}\cdot T}\right)}\right)}\right)$$ (2.1)

N_N = N_D - N_A. (2.2)

Here N_A and N_D are the acceptor and donor doping concentrations, respectively, n_i is the intrinsic carrier concentration, and $\ensuremath{E_{\mathrm{g}}}$ is the band gap energy. The band bending causes all energy levels in the semiconductor to shift, not only the band edges. This situation is depicted for the vicinity of a pn-junction in Fig. 2.3.

Figure 2.3: Band bending in a nonuniformly doped semiconductor.

When high energy electrons are focused on the surface of a semiconductor and a considerable number of electron-hole pairs are generated, the energy spectrum of the emitted secondary electrons is characteristic for the semiconductor material. Band bending in the semiconductor does not change the spectrum except for a shift of the whole spectrum. For measurements the shift of a suitable peak in the spectrum which corresponds to an energy level E_k is observed. By comparing several spectra which were measured at locations of the semiconductor with different net doping with respect to a constant reference energy E_ref the shift of each of the spectra can be determined.

The observed position dependent energy shift is directly proportional to the change of the potential at the semiconductor surface. Because of the equivalence of the energy shift and the change of the surface potential, only the potential is considered in the following discussion as this is the quantity which is used by MINIMOS-NT.

Without the injection of electrons and without the application of potentials to contacts which would cause a current flow the net doping can be calculated from the surface potential by the Poisson equation. Under the assumption that the charge of the free carriers is much smaller than the charge of the ionized dopants, which is true in a space charge region, this can be written as

$${\frac{\partial ^{2}\ensuremath{\varphi}}{\partial x^{2}}} {\frac{1}{\varepsilon_{\mathrm{0}}\cdot\varepsilon_{\mathrm{r}}}} \left(\vphantom{N_{\mathrm{D}}(x) - N_{\mathrm{A}}(x)}\right. \left.\vphantom{N_{\mathrm{D}}(x) - N_{\mathrm{A}}(x)}\right)$$ (2.3)

In this case the pn-junction is located where the second derivative of the surface potential equals zero.

The injection of primary electrons and the generation of secondary electron-hole pairs cause an additional change of the surface potential in the vicinity of the injection location (see Fig. 2.4). Therefore the net doping cannot be calculated directly from the Poisson equation (2.3). The pn-junction is no longer located at the position where the second derivative of the measured surface potential equals zero and some inverse modeling is necessary to determine the location of the pn-junction. The dependence of the observed surface potential and the net doping is investigated in detail in the following section.

Figure 2.4: The injected electrons and the electron-hole pairs generated by impact ionization cause an additional change in the surface potential in the vicinity of the injection position.