Resistivity: 4-probe measurement ELEC-L3211: Postgraduate Course in Micro and Nanoscience Libin Wang: libin.wang@aalto.fi 27/10/2016
My research experience (Master thesis) Mo2C superconducting crystal InAs semiconductor nanowire 2.0 80 K 1.8 G (e2/h) 40 K 30 K 25 K 20 K 15 K 10 K 8K 5K 1.9 K 1.6 1.4 Grown by MBE with Ag as catalyst 2 4 B (T) 6 8 Weak localization Weak anti-localization UCFs grown Chemical Vapor Deposition(CVD) Postgraduate Course in Micro and Nanoscience: Resistivity 1/22
Motivation Easy for me as I already did resistance measurement before Postgraduate Course in Micro and Nanoscience: Resistivity 2/22
Motivation Important for staring materials as it contributes to the many properties: series resistance capacitance threshold voltage hot carrier degradation of MOS device latch up of CMOS circuits This is really 4-point probe! Postgraduate Course in Micro and Nanoscience: Resistivity 3/22
introduction R = ρ L A = ρ L Wt Ohm s Law: R = V I J = σe(differential form) σ = q(nμ n + pμ p ) ρ: resistivity A=W*t: cross-section area L: Length Postgraduate Course in Micro and Nanoscience: Resistivity 4/22
Two point vs Four-point Probe R measure = 2R W + 2R C + R
Is this 4-points probe measurement? V V R I R I I No a I Yes b Postgraduate Course in Micro and Nanoscience: Resistivity 6/22
What we are measuring? Postgraduate Course in Micro and Nanoscience: Resistivity 7/22
4-point probe measurement E = Jρ = dv dr Bulk sample(3d): infinity in dimension(compared to S) Current density J = I 2πr 2 Voltage at point P: 0 VdV = Iρ r dr 2π 0 r 2 V = Iρ 2πr Voltage at P: V = Iρ 2πr = Iρ Iρ = Iρ 2πr 1 2πr 2 2π ( 1 1 ) r 1 r 2 Minus sign comes from current direction Postgraduate Course in Micro and Nanoscience: Resistivity 8/22
4-point probe measurement V 2 = Iρ 2π ( 1 s 1 1 s 2 + s 3 ) V 3 = Iρ 2π ( 1 s 1 + s 2 1 s 3 ) V = V 2 V 3 = Iρ 2π ( 1 s 1 1 s 2 + s 3 1 s 1 + s 2 + 1 s 3 ) ρ = 2π 1 s 1 1 (s 2 +s 3 1 (s 1 +s 2 ) + V 1 s 3 I (Correct equation 1.9, wrong printing on book 3 rd edition) Typical probe radii: 30-500μm Probe spacing: 0.5 1.5mm For most 4-point probe, s 1 =s 2 =s 3 =s ρ = 2πs V I Postgraduate Course in Micro and Nanoscience: Resistivity 9/22
4-point probe measurement dr = ρ dr A dx :differential length A = 2πr 2 : surface area penetrated by current R = r2 ρ dr 2πr 2 = 1 ρ 2s 2π r1 Postgraduate Course in Micro and Nanoscience: Resistivity 10/22
Correction factor Semiconductor wafer are not semi-infinite in lateral and vertical dimension ρ = 2πsF V I F = F 1 F 2 F 3 F 1 : corrects for sample thickness F 2 : corrects for lateral sample dimensions F 3 : correct for placement of the probes relative to sample edge All the correction factor we talk about next is only for collinear probes with equal probe spacing s Postgraduate Course in Micro and Nanoscience: Resistivity 11/22
Correction factor F 1 ρ = 2πsF V I For non-conducting bottom wafer surface boundary J = I 2πr 2 J = I (t s) 2πrt t: thickness of sample For conducting bottom wafer surface boundary Postgraduate Course in Micro and Nanoscience: Resistivity 12/22
Correction factor F 1 ρ = 2πsF V I For non-conducting bottom wafer surface boundary For conducting bottom wafer surface boundary Postgraduate Course in Micro and Nanoscience: Resistivity 13/22
Correction factor F 1 ρ = 2πsF V I For non-conducting bottom wafer surface boundary For t s Postgraduate Course in Micro and Nanoscience: Resistivity 14/22
Correction factor F 1 Sheet resistance R s is in units of Ohm/square. Resistance of a conductor line can now be easily calculated by breaking down the conductor into n squares: R = nr s Postgraduate Course in Micro and Nanoscience: Resistivity 15/22
Correction factor F 2 Sample size correction factor Sample must have a diameter D 40s for F 2 to be unity Postgraduate Course in Micro and Nanoscience: Resistivity 16/22
Correction factor F 2 Sample size correction factor Postgraduate Course in Micro and Nanoscience: Resistivity 17/22
Correction factor F 3 correct for placement of the probes relative to sample edge Probe distance from the wafer boundary at least 3-4 times of probe spacing Postgraduate Course in Micro and Nanoscience: Resistivity 18/22
Correction factor F 3 correct for placement of the probes relative to sample edge V 2 = Iρ 2π ( 1 s 1 1 s 2 + s 3 )(d s) Postgraduate Course in Micro and Nanoscience: Resistivity 19/22
Sample with arbitrarily shape Van der Pauw s theory: 1) The contacts are at the circumference of the sample 2) The contacts are sufficiently small 3) The sample is uniformly thick 4) Surface of sample is singly connected (i.e., samples does not contain isolated holes) F is not correction factor, F is a function depend on ratio Postgraduate Course in Micro and Nanoscience: Resistivity 20/22
Measurement errors, strengths and weaknesses errors 1. Sample Size 2. Minority/Majority Carrier Injection 3. Probe Spacing 4. Current 5. Temperature 6. Surface Preparation Weaknesses: surface damage and metal deposits on the sample Strength: popular used for long time and without calibrated Postgraduate Course in Micro and Nanoscience: Resistivity 21/22
Homework: modified with Problem 1.4 Derive an expression for resistivity of a 2D semiconductor sample with infinite in extent laterally measured with square four-point probe with the probes spaced a distance s shown below. The thickness of the sample is t, and t s. Current I enter probe 1 and leaves probe 4, voltage V is measured between probes 2 and 3. Postgraduate Course in Micro and Nanoscience: Resistivity 22/22