Four-Point Probe Measurement: Carrier Concentration & Doping Concentration in Silicon Wafer

(i) The procedure of using the four-point probe to measure a sample\u0027s carrier concentration is as follows:\n\n1. Set up the four-point probe system: Place four small probes evenly spaced apart on the surface of the sample. The spacing between the probes should be known and kept constant.\n\n2. Apply a known current through the outer two probes: Use a current source to apply a constant current between the outer two probes. This current should be small enough to avoid heating the sample.\n\n3. Measure the voltage across the inner two probes: Use a voltmeter to measure the voltage drop across the inner two probes. This voltage is directly related to the resistance of the sample.\n\n4. Calculate the sheet resistance: Use Ohm\u0027s law to calculate the sheet resistance of the sample using the measured voltage and the known current. The sheet resistance is given by R = V/I, where R is the sheet resistance, V is the voltage drop, and I is the current.\n\n5. Calculate the carrier concentration: Use the sheet resistance and the sample\u0027s thickness to calculate the carrier concentration using the formula n = 1/(qRt), where n is the carrier concentration, q is the charge of an electron, R is the sheet resistance, and t is the sample\u0027s thickness.\n\n(ii) To measure the doping concentration of a 2 inch diameter Si wafer using the four-point probe, we can follow the procedure described above.\n\nGiven:\n- Applied voltage: 20mV\n- Current measured: 10mA\n- Probe spacing: 0.5mm\n- Sample thickness: 400\u03bcm\n- Carrier mobility: 1000 cm^2/V.s\n\n1. Calculate the sheet resistance:\nUsing the measured current and voltage, we can calculate the sheet resistance using the formula R = V/I.\nR = (20mV)/(10mA) = 2\u03a9\n\n2. Calculate the carrier concentration:\nUsing the sheet resistance, thickness, and carrier mobility, we can calculate the carrier concentration using the formula n = 1/(qRt).\nGiven that the charge of an electron q is 1.6 x 10^-19 C, and the thickness t is 400\u03bcm (or 0.4mm):\nn = 1/(1.6 x 10^-19 C * 2\u03a9 * 0.4mm) = 1.56 x 10^18 cm^-3\n\nTherefore, the doping concentration of the Si wafer is approximately 1.56 x 10^18 cm^-3.