2.1.1 The AVC Potential

During an AVC measurement the extraction position x and the injection position $\xi$ are varied. These are the location of the extraction of the surface potential and the location of the injection of electrons, respectively. Therefore the AVC potential $\varphi_{\mathrm{AVC}}^{}$ is a function of the two variables x and $\xi$ .

For an ideal AVC measurement the injection and extraction location are equal and they are changed simultaneously. This position is moved along the diagonal z of the x- $\xi$ plane.

$\left.\vphantom{\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial z^{2}}}\right.$ ${\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial z^{2}}}$ $\left.\vphantom{\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial z^{2}}}\right\vert _{z = x = \xi}^{}$

(2.5)

In an actual measurement the secondary electrons are emitted from a region with a diameter of the order of the diameter of the probing primary electron beam and which extends several nanometers into the semiconductor in vertical direction.

The AVC potential is assumed to be sufficiently often continuous differentiable with respect to the variables x and $\xi$ so that we can write

${\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial z^{2}}}$ = ${\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial x^{2}}}$ + 2^. ${\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x,\xi)}{\partial x\ \partial\xi}}$ + ${\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial \xi^{2}}}$ .

(2.7)

In the following the right side of (2.7) is investigated and approximations for the three terms of the sum are derived.

The first term of the sum on the right side of (2.7) describes variations due to a change of the potential extraction position with fixed injection position. Therefore it is given by the Poisson equation in which the charge density is the sum of the charge density caused by the injection of electrons $\rho_{\mathrm{inj}}^{}$ and the charge density caused by the doping $\rho_{\mathrm{dop}}^{}$ .

${\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial x^{2}}}$ = - ${\frac{1}{\varepsilon_{\mathrm{0}}\cdot\varepsilon_{\mathrm{r}}}}$ ^. $\left(\vphantom{\rho_{\mathrm{inj}}(x,\xi) + \rho_{\mathrm{dop}}(x)}\right.$ $\rho_{\mathrm{inj}}^{}$ (x, $\xi$ ) + $\rho_{\mathrm{dop}}^{}$ (x) $\left.\vphantom{\rho_{\mathrm{inj}}(x,\xi) + \rho_{\mathrm{dop}}(x)}\right)$

(2.8)

$\displaystyle {\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial x\ \partial\xi}}$	= $\displaystyle {\frac{\partial }{\partial \xi}}$ $\displaystyle \left(\vphantom{\frac{\partial \varphi_{\mathrm{AVC}}(x,\xi)}{\partial x}}\right.$ $\displaystyle {\frac{\partial \varphi_{\mathrm{AVC}}(x,\xi)}{\partial x}}$ $\displaystyle \left.\vphantom{\frac{\partial \varphi_{\mathrm{AVC}}(x,\xi)}{\partial x}}\right)$
	= $\displaystyle {\frac{\partial }{\partial \xi}}$ $\displaystyle \left(\vphantom{\int\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial x^{2}}\ dx}\right.$ $\displaystyle \int$ $\displaystyle {\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial x^{2}}}$ dx $\displaystyle \left.\vphantom{\int\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial x^{2}}\ dx}\right)$ .

(2.9)

${\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x,\xi)}{\partial x\ \partial\xi}}$ = - ${\frac{1}{\varepsilon_{\mathrm{0}}\cdot\varepsilon_{\mathrm{r}}}}$ ^. ${\frac{\partial }{\partial \xi}}$ $\left(\vphantom{\int\rho_{\mathrm{inj}}(x, \xi)\ dx + \int\rho_{\mathrm{dop}}(x)\ dx}\right.$ $\int$ $\rho_{\mathrm{inj}}^{}$ (x, $\xi$ ) dx + $\int$ $\rho_{\mathrm{dop}}^{}$ (x) dx $\left.\vphantom{\int\rho_{\mathrm{inj}}(x, \xi)\ dx + \int\rho_{\mathrm{dop}}(x)\ dx}\right)$

(2.10)

The second term of the sum on the right side of (2.10) vanishes because the integrand does not dependent on the injection position $\xi$ .

${\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x,\xi)}{\partial x\ \partial\xi}}$ = - ${\frac{1}{\varepsilon_{\mathrm{0}}\cdot\varepsilon_{\mathrm{r}}}}$ ^. ${\frac{\partial }{\partial \xi}}$ $\left(\vphantom{\int\rho_{\mathrm{inj}}(x, \xi)\ dx}\right.$ $\int$ $\rho_{\mathrm{inj}}^{}$ (x, $\xi$ ) dx $\left.\vphantom{\int\rho_{\mathrm{inj}}(x, \xi)\ dx}\right)$

(2.11)

For uniform doping there is no difference between moving the injection position $\xi$ and varying the extraction location x by the same amount in the opposite direction.

${\frac{\partial }{\partial \xi}}$ $\left(\vphantom{\int\rho_{\mathrm{inj}}(x, \xi)\ dx}\right.$ $\int$ $\rho_{\mathrm{inj}}^{}$ (x, $\xi$ ) dx $\left.\vphantom{\int\rho_{\mathrm{inj}}(x, \xi)\ dx}\right)$ = - ${\frac{\partial }{\partial x}}$ $\left(\vphantom{\int\rho_{\mathrm{inj}}(x, \xi)\ dx}\right.$ $\int$ $\rho_{\mathrm{inj}}^{}$ (x, $\xi$ ) dx $\left.\vphantom{\int\rho_{\mathrm{inj}}(x, \xi)\ dx}\right)$

(2.12)

$\displaystyle {\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial x\ \partial\xi}}$

= - $\displaystyle {\frac{1}{\varepsilon_{\mathrm{0}}\cdot\varepsilon_{\mathrm{r}}}}$ ^. $\displaystyle \left(\vphantom{-\frac{\partial }{\partial x}\left(\int\rho_{\mathrm{inj}}^{0}(x, \xi)\ dx\right)}\right.$ - $\displaystyle {\frac{\partial }{\partial x}}$ $\displaystyle \left(\vphantom{\int\rho_{\mathrm{inj}}^{0}(x, \xi)\ dx}\right.$ $\displaystyle \int$ $\displaystyle \rho_{\mathrm{inj}}^{0}$ (x, $\displaystyle \xi$ ) dx $\displaystyle \left.\vphantom{\int\rho_{\mathrm{inj}}^{0}(x, \xi)\ dx}\right)$ $\displaystyle \left.\vphantom{-\frac{\partial }{\partial x}\left(\int\rho_{\mathrm{inj}}^{0}(x, \xi)\ dx\right)}\right)$

= - $\displaystyle {\frac{1}{\varepsilon_{\mathrm{0}}\cdot\varepsilon_{\mathrm{r}}}}$ ^. $\displaystyle \left(\vphantom{-\rho_{\mathrm{inj}}^{0}(x, \xi)}\right.$ - $\displaystyle \rho_{\mathrm{inj}}^{0}$ (x, $\displaystyle \xi$ ) $\displaystyle \left.\vphantom{-\rho_{\mathrm{inj}}^{0}(x, \xi)}\right)$ ,

(2.13)

where $\rho_{\mathrm{inj}}^{0}$ is the charge density caused by injection for a uniform doping. For nonuniform doping (2.11) can be written like (2.13) with the additional term $\rho_{\mathrm{inj}}^{\mathrm{a}}$ in which all effects of the nonuniform doping are summarized.

${\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x,\xi)}{\partial x\ \partial\xi}}$ = - ${\frac{1}{\varepsilon_{\mathrm{0}}\cdot\varepsilon_{\mathrm{r}}}}$ ^. $\left(\vphantom{-\rho_{\mathrm{inj}}^{0}(x, \xi) + \rho_{\mathrm{inj}}^{\mathrm{a}}(x, \xi)}\right.$ - $\rho_{\mathrm{inj}}^{0}$ (x, $\xi$ ) + $\rho_{\mathrm{inj}}^{\mathrm{a}}$ (x, $\xi$ ) $\left.\vphantom{-\rho_{\mathrm{inj}}^{0}(x, \xi) + \rho_{\mathrm{inj}}^{\mathrm{a}}(x, \xi)}\right)$

(2.14)

The third term of the sum on the right side of (2.7) describes the effect of varying the injection position $\xi$ and keeping the extraction position x constant. For a first approximation again constant doping is assumed. Then for the second derivative of the AVC potential there is no contribution from the charge caused by the doping $\rho_{\mathrm{dop}}^{}$ . A change of the injection position has the same effect on the AVC potential as a change of the extraction position by the same amount in the same direction.

For uniform doping the third term of the sum on the right side of (2.7) can then be written as

$\displaystyle {\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial \xi^{2}}}$	= $\displaystyle \left.\vphantom{\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial x^{2}}}\right.$ $\displaystyle {\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial x^{2}}}$ $\displaystyle \left.\vphantom{\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial x^{2}}}\right\vert _{\rho_{\mathrm{dop}}\equiv 0}^{}$
	= - $\displaystyle {\frac{1}{\varepsilon_{\mathrm{0}}\cdot\varepsilon_{\mathrm{r}}}}$ ^. $\displaystyle \rho_{\mathrm{inj}}^{0}$ (x, $\displaystyle \xi$ ).

(2.16)

Similarly to the approximation of the second term of (2.7) the effects of nonuniform doping are summarized in an additional term $\rho_{\mathrm{inj}}^{\mathrm{b}}$ (x, $\xi$ ).

${\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial \xi^{2}}}$ = - ${\frac{1}{\varepsilon_{\mathrm{0}}\cdot\varepsilon_{\mathrm{r}}}}$ ^. $\left(\vphantom{\rho_{\mathrm{inj}}^{0}(x, \xi) + \rho_{\mathrm{inj}}^{\mathrm{b}}(x, \xi)}\right.$ $\rho_{\mathrm{inj}}^{0}$ (x, $\xi$ ) + $\rho_{\mathrm{inj}}^{\mathrm{b}}$ (x, $\xi$ ) $\left.\vphantom{\rho_{\mathrm{inj}}^{0}(x, \xi) + \rho_{\mathrm{inj}}^{\mathrm{b}}(x, \xi)}\right)$

(2.17)

Combining the approximations (2.8), (2.13), and (2.17) the second derivative of the AVC potential (2.7) finally can be written as

$\displaystyle {\frac{\partial ^{2}\varphi_{\mathrm{AVC}}(x, \xi)}{\partial z}}$

= - $\displaystyle {\frac{1}{\varepsilon_{\mathrm{0}}\cdot\varepsilon_{\mathrm{r}}}}$ ^. $\displaystyle \left(\vphantom{\rho_{\mathrm{dop}}(x) + 2\cdot\rho_{\mathrm{inj}}^{\mathrm{a}}(x, \xi) + \rho_{\mathrm{inj}}^{\mathrm{b}}(x, \xi)}\right.$ $\displaystyle \rho_{\mathrm{dop}}^{}$ (x) + 2^. $\displaystyle \rho_{\mathrm{inj}}^{\mathrm{a}}$ (x, $\displaystyle \xi$ ) + $\displaystyle \rho_{\mathrm{inj}}^{\mathrm{b}}$ (x, $\displaystyle \xi$ ) $\displaystyle \left.\vphantom{\rho_{\mathrm{dop}}(x) + 2\cdot\rho_{\mathrm{inj}}^{\mathrm{a}}(x, \xi) + \rho_{\mathrm{inj}}^{\mathrm{b}}(x, \xi)}\right)$

= - $\displaystyle {\frac{1}{\varepsilon_{\mathrm{0}}\cdot\varepsilon_{\mathrm{r}}}}$ ^. $\displaystyle \left(\vphantom{\rho_{\mathrm{dop}}(x) + \Delta\rho_{\mathrm{inj}}(x,\xi)}\right.$ $\displaystyle \rho_{\mathrm{dop}}^{}$ (x) + $\displaystyle \Delta$ $\displaystyle \rho_{\mathrm{inj}}^{}$ (x, $\displaystyle \xi$ ) $\displaystyle \left.\vphantom{\rho_{\mathrm{dop}}(x) + \Delta\rho_{\mathrm{inj}}(x,\xi)}\right)$ ,

(2.18)

The second derivative of the AVC potential $\varphi_{\mathrm{AVC}}^{}$ is proportional to the superposition of a contribution of the charge density caused by the doping and a contribution due to the injection of electrons.

To calculate $\Delta$ $\rho_{\mathrm{inj}}^{}$ one has to solve the drift-diffusion equations which has to be done by numeric integration for the general case. At the injection position the injected charge is an upper bound for $\Delta$ $\rho_{\mathrm{inj}}^{}$ .

From (2.18) it is clear that only when $\Delta$ $\rho_{\mathrm{inj}}^{}$ is several orders of magnitude smaller than $\rho_{\mathrm{dop}}^{}$ the location of the pn-junction is at the position where the second derivative of $\varphi_{\mathrm{AVC}}^{}$ equals zero and therefore it can be extracted from the measured AVC potential $\varphi_{\mathrm{AVC}}^{}$ without inverse modeling.

For uniform doping (2.18) does not dependent on the beam current. Therefore an approximation is needed only for the space charge region where the doping changes considerably.