Corresponding author: Natalya V. Latukhina (natalat@yandex.ru)

In this work we have used contact and contactless techniques to measure the electrical resistivity of single crystal silicon wafers with porous layers of variable thickness synthesized on the surface. The porous layers have been synthesized on the surfaces of single crystal wafers with well pronounced microroughness pattern, either textured or grinded. We have used the classic four-probe method with a linear probe arrangement as the contact measurement technique, and the resonance microwave method based on microwave absorption by free carriers as the contactless measurement technique. Electrical resistivity distribution over the specimen surface has been mapped based on the measurement results. We have demonstrated a general agreement between the electrical resistivity distribution patterns as measured using the contact and contactless measurement techniques. To analyze the electrical resistivity scatter over the specimen surface area we have simulated the field distribution in the electrolyte during porous layer formation in a non-planar anode cell. The regularities of the electrical resistivity spatial distribution in different types of specimens are accounted for by specific porosity formation mechanisms which in turn are controlled by the initial microroughness pattern and the field distribution pattern in the electrolyte for each specific case.

Contactless parameter measurement techniques are of special interest for nanomaterials which include porous silicon because contact measurement of their parameters may cause irreversible damage to their nanostructure. An urgent problem is the treatise of nanomaterial parameter contactless measurement results and their comparison with conventional contact technique data.

Electrical resistivity of porous silicon may very over an extremely wide range [^{3}^{5} compounds) [

The porous silicon layer was synthesized on the wafer surface by electrochemical etching in hydrofluoric acid alcohol/water solutions in vertical electrolytic cells. We used

The electrical resistivity of the specimens was measured with two techniques, i.e., the classic four-probe method with a linear probe arrangement and, as the contactless measurement technique, the resonance microwave method based on microwave absorption by free carriers for it allows measurements to be carried out without introducing contamination or damaging the surface structure of the specimens. The four-probe method was used for calibrating the contactless method.

The operation principle of the BKI-UES contactless measurement instrument is as follows. A

The contactless microwave method is a calibrated one, i.e., the output signal expressed in relative units can be converted to electrical resistivity only by comparing with results for similarly shaped specimens having known electrical resistivity. For this experiment we measured the specimens simultaneously with the contactless and the four-probe techniques in order to immediately convert the output signal to electrical resistivity units. This eliminated the necessity of using reference specimens.

Four-probe measurements were carried out with a VIK-UES-A instrument at 80 points with a 2 mm steps over the area of a circle inscribed into a square wafer. Series of three measurements for each wafer were taken for average value calculation at each point and plotting a color 3D map of electrical resistivity distribution over the specimen surface which illustrated its inhomogeneity. Contactless measurements were carried out with 5 mm steps at 20 points along the quadrate diagonal.

This work was aimed at simulating the field distribution in the electrolyte at the silicon/electrolyte boundary and comparing it with the electrical resistivity distribution maps. We studied the 2D electric fields by analyzing the potential distribution along planes perpendicular to the electrodes. Before an external current source is switched on, an electrode potential exists in the system due to the double electric field at the semiconductor/electrolyte boundary: the polarized molecules of the solution cause ions in the surface semiconductor layer to hydrate and transfer to the solution thus charging it positively, while the excess electrons in the semiconductor produce a negative charge. The negative charge at the electrode prevents cation transfer to the solution while part of the cations in the solution interact with electrons and enter the sites of the crystal lattice they left. A dynamic equilibrium is established when the cation emission and return rates equalize. This results in the generation of a double electric layer similar to a flat capacitance one plate of which is the semiconductor surface and the other is the layer of ions in the electrolyte solution. The electrode potential as a function of the cation concentration in the solution and the temperature is described by the Nernst equation:

where φ^{0} is the standard electrode potential,

Once an external voltage source is connected to the system, current starts passing through the electrolytic bath resulting in a shift of the potentials from the equilibrium values, i.e., to electrode polarization. The chemical polarization of the electrodes produces a voltaic cell with the electromotive force direction opposite to that of the external electromotive force. Therefore the voltage of electrolysis is generally the sum of the polarization electromotive force, the anodic and cathodic overvoltages and the ohmic potential drop at the electrolyte. For the case in question the electrolysis reaction at the electrode/electrolyte boundary is so intense that the electrode kinetics can be neglected and the barrier potential difference deviates from its equilibrium value but slightly. In other words there is no activation overvoltage and hence the current distribution only depends on the anode and cathode shapes. For the textured surface specimens the anode was a regular array of similar regular triangles and for the grinded surface specimens, an irregular array of differently sized triangles.

The field was computer simulated using the COMSOL Multiphysics software package (for electrochemical cell simulation). The boundary conditions were as follows: the electrolyte was considered electrically neutral uncompressible liquid with negligible composition variation and no turbulence,

The main equations used for the computer model were as follows: diffusion current as a function of ion concentration and electrical field magnitude in the electrolyte was described using the Nernst–Plank equation:

where _{m}^{-1}cm^{-2}; ^{2}s^{-1}; ^{-3}; _{m}_{m}^{-2}mol×s^{-1}Cl^{-1}V^{-1}; ^{-1} (_{a}); field magnitude expressed via potential gradient.

The current density in the electrolyte is

The current density at the electrode is

where δ_{l}_{s}

The calculation results were presented in the form of current density vector field map in the plane perpendicular to the wafer surface. Since the electric field and the current density are in simple relationships such as Eqs. (3)–(4), the current density vector field maps will be identical to the electrolyte field distribution maps.

The four-probe electrical resistivity measurement results are summarized in Table

Four-probe technique electrical resistivity distribution over textured specimen surface: (_{av} = 2±0.13 Ohm×cm; (_{av} = 3.2±0.11 Ohm×cm; (_{av} = 2.13±0.09 Ohm×cm; (_{av} = 3.1±0.5 Ohm×cm.

Contactless technique electrical resistivity distribution along quadrate diagonal for textured specimens with (

(arrows) field magnitude and (color) potential distribution in electrolyte for textured specimen etching. Left-hand digital

Specimen parameters as measured by four-probe technique.

Specimen # | Surface type | ρ_{max}, Ohm·cm |
ρ_{min}, Ohm·cm |
ρ_{av}, Ohm·cm |
(ρ_{max} – ρ_{min})/ρ_{av.} |
RMS Δρ, Ohm·cm | Δρ/ρ_{av.}, % |
---|---|---|---|---|---|---|---|

T0 | Initial textured | 2.3 | 1.3 | 2 | 0.5 | 0.13 | 7 |

T5 | Textured with porous layer, 5 min etching | 3.5 | 2.8 | 3.2 | 0.22 | 0.12 | 4 |

T10 | Textured with porous layer, 10 min etching | 2.4 | 1.95 | 2.1 | 0.21 | 0.09 | 5 |

Sh0 | Initial grinded | 2.1 | 1.3 | 1.6 | 0.57 | 0.12 | 8 |

Sh1 | Grinded with porous layer, 5 min etching | 2.5 | 1.8 | 2.2 | 0.32 | 0.13 | 6 |

Sh2 | Grinded with porous layer, 10 min etching | 2.23 | 1. 01 | 1.9 | 0.63 | 0.15 | 8 |

Sh3 | Grinded with porous layer, 15 min etching | 4.00 | 1.8 | 2.3 | 0.96 | 0.4 | 17 |

Sh3 (fragm.) | Grinded with porous layer, 15 min etching | 4.00 | 3.4 | 2.3 | 0.26 | – | – |

Specimen parameters as measured by contactless technique.

Specimen # | Surface type | ρ_{max}, Ohm·cm |
ρ_{min}, Ohm·cm |
ρ_{av}, Ohm·cm |
(ρ_{max} – ρ_{min})/ρ_{av.} |
RMS Δρ, Ohm·cm | Δρ/ρ_{av.}, %_{av} |
---|---|---|---|---|---|---|---|

2 T | Textured with porous layer, 10 min etching | 3.5 | 2.2 | 2.8 | 0.48 | 0.5 | 15 |

9T | Textured with porous layer, 15 min etching | 2.9 | 1.7 | 2.2 | 0.5 | 0.3 | 14 |

Sh1 | Grinded with porous layer, 10 min etching | 4.6 | 1.5 | 1.8 | 1.7 | 0.4 | 19 |

Sh1 (w/o spike) | Grinded with porous layer, 10 min etching | 1.9 | 1.5 | 1.7 | 0.25 | 0.14 | 8 |

Analysis of the results for specimens with grinded surfaces (Fig. _{max} – ρ_{min})/ρ_{av} = 0.57 for textured surface, (ρ_{max} – ρ_{min})/ρ_{av} = 0.5 for grinded surface, and RMS deviation is 8% for grinded surface and 7% for textured surface. However the distribution of regions with different electrical resistivity over grinded specimen surface is inhomogeneous unlike the homogeneous distribution for textured specimens, and this “edge effect” cannot be attributed to gravity as for etched specimens.

Four-probe technique electrical resistivity distribution over grinded specimen surface: (_{av} = 1.57±0.11 Ohm×cm; (_{av} = 1.91±0.14 Ohm×cm; (_{av} = 2.18±0.13 Ohm×cm; (_{av} = 2.3±0.4 Ohm×cm.

The homogeneity degrees of the grinded specimens with porous layers and specimens with the porous layers on textured surfaces differ for similar etching time. After 5 min etching both specimen types exhibit an increase in the homogeneity with an increase in the average resistivity. For 10 min etching both these parameters increase in the specimens with etched surfaces unlike those with textured surfaces. For this etching mode the etched surfaces have no more small inhomogeneities and pores form intensely in microroughness cavities. For the specimens with porous silicon layers on grinded surfaces after 15 min etching the homogeneity increases, the difference between the highest and the lowest electrical resistivity being the smallest (except for a small spike region where the electrical resistivity differs largely from that in other regions) which is not the case for the specimens with textured surfaces. This behavior can be accounted for by field distribution regularities in the electrolyte for grinded anode surface (Fig.

Contactless technique electrical resistivity distribution along quadrate diagonal for grinded specimens with 10 min etched porous silicon layer.

Comparison of the electrical resistivity distributions for contact four-probe method and along the diagonal of the specimens according to the contactless technique (Fig.

(arrows) current density and (color) potential distribution in electrolyte for grinded specimen etching.

Data comparison between contact and contactless techniques.

Specimen # | Contact / Contactless | Surface type | ρ_{max}, Ohm·cm |
ρ_{min}, Ohm·cm |
ρ_{av}, Ohm·cm |
(ρ_{max} –ρ_{min})/ρ_{av.} |
RMS Δρ, Ohm·cm | Δρ/ρ_{av.}, % |
---|---|---|---|---|---|---|---|---|

2T | CL | Textured with porous layer, 10 min etching | 3.5 | 2.1 | 2.8 | 0.5 | 0.7 | 25 |

T10 | C | Textured with porous layer, 10 min etching | 2.4 | 1.95 | 2.1 | 0.21 | 40 | |

9T | CL | Textured with porous layer, 15 min etching | 2.9 | 1.7 | 2.2 | 0.5 | 1 | 50 |

T7 | C | Textured with porous layer, 15 min etching | 4 | 2 | 3 | 0.67 | 30 | |

Sh2 | C | Grinded with porous layer, 10 min etching | 2.2 | 1.1 | 1.9 | 0.63 | 0.24 | 13 |

Sh1 (w/o spike) | CL | Grinded with porous layer, 10 min etching | 1.9 | 1.5 | 1.7 | 0.13 | 15 |

Analysis of the results leads to the following practically useful conclusions.

The electrical resistivity distribution over porous silicon surface depends on the initial surface microroughness and electrolytical etching time;

Field distribution simulation for electrochemical etching suggests that at the silicon anode/electrolyte boundary it is controlled by the initial surface microroughness;

Taking into account the inhomogeneity of the initial materials and the difference in the electrical resistivity measurement techniques one can accept that the electrical resistivity distribution patterns over the wafer surface are roughly the same for contact and contactless measurements and correspond to the electrical field distribution in the electrolyte;

The numeric parameters yielded by contact and contactless measurements correlate. The difference between the average electrical resistivities of the same specimen types for contact and contactless measurements is of the same order of magnitude or smaller than the difference between the highest and the lowest electrical resistivity for the specimens.

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