**3. Results**

#### *3.1. Sheet Resistance and Determination of the Acceptor Ionization Energy*

All fabricated samples show ohmic behavior across all measurement temperatures. The sheet resistance *Rsh* was used to determine the acceptor ionization energies ∆*E<sup>A</sup>* of the fabricated samples. Equation (3) (together with Equations (4)–(8)) can be used to describe the sheet resistance *Rsh* of a semiconductor, where *q* indicates the elementary charge, *t* the thickness of the semiconductor layer, *p* and *n* the hole and electron concentrations, *µ<sup>p</sup>* and *µ<sup>n</sup>* the hole and electron mobilities, respectively.

$$R\_{\rm sh} = \left( q \int\_0^t \left( \mu\_n(\mathbf{x}) n(\mathbf{x}) + \mu\_p(\mathbf{x}) p(\mathbf{x}) \right) d\mathbf{x} \right)^{-1} \tag{3}$$

Equations (4)–(6) were used to calculate the hole and electron mobility and their respective temperature dependence, where *µconst* describes the mobility due to phonon scattering, *µdop* the doping dependent mobility degradation, *T* the temperature, *N<sup>D</sup>* the donor concentration and *N<sup>A</sup>* the acceptor concentration (all other parameters and their values are shown in Table A1 [19].

$$
\mu = \left(\mu\_{const}^{-1} + \mu\_{dop}^{-1}\right)^{-1} \tag{4}
$$

$$
\mu\_{\rm const} = \mu\_L \left(\frac{T}{300\text{K}}\right)^{-\frac{\pi}{6}}\tag{5}
$$

$$\mu\_{dop} = A\_{min} \left(\frac{T}{300\text{K}}\right)^{a\_m} + \frac{A\_d \left(\frac{T}{300\text{K}}\right)^{a\_d}}{1 + \left(\frac{N\_A + N\_D}{A\_N \left(\frac{T}{300\text{K}}\right)^{a\_N}}\right)^{A\_d \left(\frac{T}{300\text{K}}\right)^{a\_d}}}\tag{6}$$

Equations (7) and (8) were used to describe the carrier ionization, where *N* + *D* describes the ionized donor concentration and *N* − *A* describes the ionized acceptor concentration (all other parameters and their values are shown in Table A2 [1]. Here, a negligible carrier compensation (*p* ≈ *N* − *A* ) was assumed at first.

$$n \approx N\_D^+ = \frac{\eta\_n}{2} \left( \sqrt{1 + \frac{4N\_D}{\eta\_n}} - 1 \right) \text{ with } \begin{array}{l} \eta\_n = \frac{N\_C}{\mathcal{S}D} \exp\left(-\frac{\Delta E\_D}{kT}\right) \\\ N\_C = N\_{C, 300\text{K}} \left(\frac{T}{300\text{K}}\right)^{1.5} \end{array} \tag{7}$$

$$p \approx \mathcal{N}\_A^- = \frac{\eta\_p}{2} \left( \sqrt{1 + \frac{4N\_A}{\eta\_p}} - 1 \right) \text{ with } \begin{array}{c} \eta\_p = \frac{N\_V}{\mathcal{S}A} \exp\left(-\frac{\Delta E\_A}{kT}\right) \\\ N\_V = N\_{V, \text{300K}} \left(\frac{T}{500 \text{K}}\right)^{1.5} \end{array} \tag{8}$$

In Figure 1a the normalized average measured sheet resistances and the associated standard error of all sets of samples with an implanted Al surface concentration of 5 × 10 19 cm−<sup>3</sup> are shown. It can be seen that all sets of samples show similar temperature dependent behavior despite differences in sheet resistance values (see inset in Figure 1a).

) Measurement temperature dependence of normalized average sheet resistance ℎ acceptor ionization energies Δ **Figure 1.** (**a**) Measurement temperature dependence of normalized average sheet resistance *Rsh* ; (**b**) Determined effective acceptor ionization energies ∆*EA*.

 the Boltzmann constant, The associated acceptor ionization energies ∆*E<sup>A</sup>* were determined by fitting the theoretical sheet resistance to the measured ones. The theoretical sheet resistance was determined by using the simulated implantation profile and assuming 100% activation of the dopants.

ℎ <sup>∗</sup> <sup>0</sup> ∝ <sup>00</sup> � <sup>00</sup> � ℎ <sup>00</sup> ℎ π � ∗0 The determined effective acceptor ionization energies ∆*E<sup>A</sup>* as well as the theoretical acceptor ionization energies (see Equation (9) with ∆*EA*,0 = 0.265 eV [19,20]) are shown in Figure 1b. It can be seen that the determined acceptor ionization energies differ significantly from the theoretical ones, which can be explained by a significant amount of carrier compensation centers (see Equation (8)). As discussed in Section 3.3, these compensation centers might be modelled by donor-like defects that trap free holes.

a shows the normalized average

 

$$
\Delta E\_A = \Delta E\_{A,0} - \sqrt[3]{N\_D + N\_A} \tag{9}
$$

− − − �

 <sup>3</sup> <sup>−</sup>

4

 <sup>2</sup><sup>0</sup> 3 3

−3

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