HW 1_SP24

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243

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Electrical Engineering

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Apr 3, 2024

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MSE 243 Electronic Materials Characterization Spring 2024 Homework #1 Due: Monday February 12 th by 11:59 PM (submit via bcourses) 1. (20 points) Derive an expression for the resistivity of a semiconductor sample infinite in extent laterally and vertically measured with a square four-point probe with the probes spaced a distance s shown in the figure below. Current I enters probe 1 and leaves probe 4; voltage V is measured between probes 2 and 3: 2. (20 points) The resistance of the semiconductor sample in the figure below is measured between the two contacts as a function of wafer thickness t . The results are: Determine the resistivity ρ in W · cm and the specific contact resistance ρ c in W · cm 2 for d = 0.01 cm. Assume the current is confined to the area of the contact, shown by the shaded region. The contact is circular with the contact resistance given by R c = ρ c /A, where A is the contact area.

MSE 243 Electronic Materials Characterization Spring 2024 3. (20 points) An arbitrarily shaped, uniformly doped van der Pauw sample has a thickness of 500 μm. The measured resistances are R 12,34 = 90 W and R 23,41 = 9 W . Determine the resistivity and the sheet resistance of this sample. 4. (20 points) Watch this video ( https://youtu.be/SJqwGAJImzo ). In this video, a student uses of a four-point probe to measure the resistivity of a 150 nm thick gold thin film deposited on a silicon wafer. (a) Based on the measured values shown in the video, calculate the resistivity (r) and sheet resistance (r s ) of the gold thin film. (b) How does this value compare to the reported room temperature resistivity of gold? Be sure to reference your source. 5. A multi-point test structure (Figure 1 a,b) was fabricated to measure the resistivity of a 400 nm diameter p-type doped silicon nanowire. Current-voltage measurements (Figure 1c) were carried out by applying a current between contacts 2 and 5 and measuring the voltage (DV) between contacts 3 and 4. Assume that the accepters are fully ionized (p=N a ). Also assume the spacing between all contacts is 1.1 mm. (a) Calculate the resistivity of the silicon nanowire assuming the current flows through the entire wire diameter (r=r p in Figure 1d). Using the information provided in Appendix 1.1 of Chapter 1 of the textbook (also included below), estimate the acceptor concentration (N a ) in nanowire. (b) To get a more accurate resistivity value, the calculation should account for the depletion region at the nanowire surface that forms due to band bending associated with interface charge at the metal/silicon contact (Figure 1d). This reduces the effective radius (r e ) of the nanowire that carries current so the calculated resistivity must be corrected to account for the depletion region. To maintain charge neutrality, the charge at the nanowire surface due to interface states (N it ) must be equal to the space charge in the depletion region: Assuming that N it /N a =7.8x10 -7 cm, calculate the effective radius (r e ) and use this to calculate a revised value for the nanowire resistivity. Using Appendix 1.1 of Chapter 1 of the textbook (also included below), estimate the acceptor concentration (N a ) in the nanowire.

MSE 243 Electronic Materials Characterization Spring 2024 (d) Figure 1. (a) Schematic and (b) SEM image of electrical test structure for silicon nanowire resistivity measurements. The contacts are evenly spaced by 1.1 μ m and the nanowire diameter is 400 nm. (c) Voltage ( D V) measured between contacts 3 and 4 for as a function of time for several currents (I) applied between contacts 2 and 5. (d) Schematic showing depletion region that forms at nanowire surface due to band bending which reduces the effective radius of the nanowire that carries current. (ref. C. Kendrick, et al. J. Appl. Phys. 122, 235101 (2017))

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