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Lecture 17: Resistivity and Hall Effect Measurements

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Lecture 17: Resistivity and Hall Effect Measurements ECE-656: Fall 2011 Lecture 17: Resistivity and Hall effect Measurements Mark Lundstrom Purdue University West Lafayette, IN USA 10/5/11 1 measurement of conductivity / resistivity 1) Commonly-used to characterize electronic materials. 2) Results can be clouded by several effects – e.g. contacts, thermoelectric effects, etc. 3) Measurements in the absence of a magnetic field are often combined with those in the presence of a B-field. This lecture is a brief introduction to the measurement and characterization of near-equilibrium transport. 2 Lundstrom ECE-656 F11 resistivity / conductivity measurements diffusive transport assumed For uniform carrier concentrations: We generally measure resisitivity (or conductivity) because for diffusive samples, these parameters depend on material properties and not on the length of the resistor or its width or cross-sectional area. Lundstrom ECE-656 F11 3 Landauer conductance and conductivity “ideal” contacts For ballistic or quasi-ballistic transport, replace the mfp with cross-sectional the “apparent” mfp: area, A n-type semiconductor 4 (diffusive) Lundstrom ECE-656 F11 conductivity and mobility “ideal” contacts cross-sectional 1) Conductivity depends on EF. area, A 2) EF depends on carrier density. 3) So it is common to characterize n-type semiconductor the conductivity at a given carrier density. 4) Mobility is often the quantity that is quoted. So we need techniques to measure two quantities: 1) conductivity 2) carrier density 5 Lundstrom ECE-656 F11 2D: conductivity and sheet conductance 2D electrons n-type semiconductor ⎛ Wt ⎞ ⎛ W ⎞ G = σ = σ t n ⎜ L ⎟ n ⎜ L ⎟ ⎝ ⎠ ⎝ ⎠ Top view “sheet conductance” 6 Lundstrom ECE-656 F11 2D electrons vs. 3D electrons Top view n-type semiconductor 3D electrons: 2D electrons: 7 Lundstrom ECE-656 F11 mobility 1) Measure the conductivity: 2) Measure the sheet carrier density: 3) Deduce the mobility from: 4) Relate the mobility to material parameters: 8 Lundstrom ECE-656 F11 recap There are three near-equilibrium transport coefficients: conductivity, Seebeck (and Peltier) coefficient, and the electronic thermal conductivity. We can measure all three, but in this brief lecture, we will just discuss the conductivity. Conductivity depends on the location of the Fermi level, which can be set by controlling the carrier density. So we need to discuss how to measure the conductivity (or resistivity) and the carrier density. Let’s discuss the resistivity first. 9 Lundstrom ECE-656 F11 outline 1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der Pauw method 5. Summary This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/ Lundstrom ECE-656 F11 10 2-probe measurements Top view L RCH = ρS W 1 2 V21 = I 2RC + RCH ( ) V21 RCH ≠ I 11 Lundstrom ECE-656 F11 “transmission line measurements” Top view 1 2 3 4 5 X X X V = I 2R + ρ S W X ji ( C S ji ) H.H. Berger, “Models for Contacts to Planar Devices,” Solid-State Electron., 15, 145-158, 1972. 12 Lundstrom ECE-656 F11 transmission line measurements (TLM) 1 2 3 4 5 Side view i j 13 Lundstrom ECE-656 F11 contact resistance (vertical flow) metal contact Area = AC interfacial layer n-Si Top view Side view 14 Lundstrom ECE-656 F11 contact resistance (vertical flow) 10−8 < ρ < 10−6 Ω-cm2 C “interfacial contact resistivity” interfacial A = 0.10 µm × 1.0µm C layer ρ = 10−7 Ω-cm2 n-Si C R = 100 Ω C Side view 15 Lundstrom ECE-656 F11 contact resistance (vertical + lateral flow) ρ L “transfer length” L = C cm T T ρ SD (W into page) 16 Lundstrom ECE-656 F11 contact resistance 17 Lundstrom ECE-656 F11 transfer length measurments (TLM) 1 2 3 4 5 Side view X X X X 1) Slope gives sheet resistance, intercept gives contact resistance 2) Determine specific contact resistivity and transfer length: 18 four probe measurements 1 2 3 4 Side view 1) force a current through probes 1 and 4 2) with a high impedance voltmeter, measure the voltage between probes 2 and 3 (no series resistance) 19 Lundstrom ECE-656 F11 Hall bar geometry pattern created with Top view thin film photolithography isolated from substrate 1 2 0 5 3 4 (high impedance voltmeter) no contact resistance Contacts 0 and 5: “current probes” Contacts 1 and 2 (3 and 4): “voltage probes” 20 Lundstrom ECE-656 F11 outline 1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der Pauw method 5. Summary This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/ Lundstrom ECE-656 F11 21 Hall effect current in x-direction: n-type semiconductor B-field in z-direction: Hall voltage measured in the y-direction: The Hall effect was discovered by Edwin Hall in 1879 and is widely used to characterize22 electronic materials.Lundstrom It also ECE-656 finds F11 use magnetic field sensors.22 Hall effect: analysis Top view of a 2D film + + + + + + + + + + n-type “Hall factor” - - - - - - - - - - - - - J nq - r E B n = µnE (σ nµn H ) × “Hall concentration” 23 23 Lundstrom ECE-656 F11 example Top view 1 2 0 5 3 4 B = 0: What are the: 1) resistivity? 2) sheet carrier density? B ≠ 0: 3) mobility? 24 Lundstrom ECE-656 F11 example: resistivity Top view 1 2 0 5 3 4 B = 0: resistivity: B = 0.2T: 25 Lundstrom ECE-656 F11 example: sheet carrier density Top view 1 2 0 5 3 4 B = 0: sheet carrier density: B = 0.2T: 26 Lundstrom ECE-656 F11 example: mobility Top view 1 2 0 5 3 4 B = 0: mobility: B = 0.2T: 27 Lundstrom ECE-656 F11 re-cap Top view 1) Hall coefficient: 1 2 0 5 3 4 2) Hall concentration: 3) Hall mobility: 4) Hall factor: 28 outline 1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der Pauw method 5. Summary This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/ Lundstrom ECE-656 F11 29 van der Pauw sample 2D film Top view arbitrarily shaped homogeneous, isotropic (no holes) M P N O Four small contacts along the perimeter Lundstrom ECE-656 F11 30 van der Pauw approach Resistivity Hall effect B-field = 0 B-field M P M P N O N O 1) force a current in M and out N 1) force a current in M and out O 2) measure VPO 2) measure VPN 3) RMN, OP = VPO / I related to ρS 3) RMO, NP = VPN / I related to VH Lundstrom ECE-656 F11 31 van der Pauw approach: Hall effect Hall effect J = σ E - σ µ r E B x n x ( n n H ) y z J = σ E + σ µ r E B y n y ( n n H ) x z M P E = ρ J + ρ µ B J x n x ( n H z ) y O N E y = − ρnµH Bz J x + ρn J y ( ) P P V B = − •dl = − dx + dy PN ( z ) ∫E ∫E x E y N N 1 J = σ - σ µ r E × B V ≡ ⎡V +B − V −B ⎤ n nE ( n n H ) H ⎣ PN ( z ) PN ( z )⎦ 2 Lundstrom ECE-656 F11 32 van der Pauw approach: Hall effect Hall effect y x ⎡ P P ⎤ V = ρ µ B ⎢ J dy − J dx⎥ H n H z ∫ x ∫ y ⎢ y x ⎥ ⎣ N N ⎦ M P P I = ∫ Jinˆ dl V = ρ µ B I N H n H z N O So we can do Hall effect measurements on such samples. For the missing steps, see Lundstrom, J = nqµ E - σ µ r E × B Fundamentals of Carrier Transport, 2nd Ed., n n ( n n H ) Sec. 4.7.1. 33 Lundstrom ECE-656 F11 van der Pauw approach: resistivity Resistivity semi-infinite half-plane M P N O M N O P 34 Lundstrom ECE-656 F11 van der Pauw approach: resistivity semi-infinite half-plane M 35 Lundstrom ECE-656 F11 van der Pauw approach: resistivity semi-infinite half-plane M N O P but there is also a contribution from contact N 36 Lundstrom ECE-656 F11 van der Pauw approach: resistivity semi-infinite half-plane M N O P it can be shown that: Given two measurements of resistance, this equation can be solved for the sheet resistance. 37 Lundstrom ECE-656 F11 van der Pauw approach: resistivity semi-infinite half-plane M P M N O P N O The same equation applies for an arbitrarily shaped sample! 38 Lundstrom ECE-656 F11 van der Pauw technique: regular sample Top view M P N O Force I through two contacts, measure V between the other two contacts. 39 Lundstrom ECE-656 F11 van der Pauw technique: summary M P M P 1) measure nH N O N O B-field B-field X 40 Lundstrom ECE-656 F11 van der Pauw technique: summary B = 0 M P M P 2) measure ρS 3) determine µ N O H N O 41 Lundstrom ECE-656 F11 outline 1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der Pauw method 5. Summary This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/ Lundstrom ECE-656 F11 42 summary 1) Hall bar or van der Pauw geometries allow measurement of both resistivity and Hall concentration from which the Hall mobility can be deduced.
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