Lecture 17: Resistivity and Hall effect Measurements

Transcription

1 ECE-656: Fall 2011 Lecture 17: Resistivity and Hall effect easurements ark Lundstrom urdue University West Lafayette, I 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) easurements 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

2 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. 3 Landauer conductance and conductivity cross-sectional area, A ideal contacts For ballistic or quasi-ballistic transport, replace the mfp with the apparent mfp: n-type semiconductor 4 (diffusive)

3 conductivity and mobility cross-sectional area, A ideal contacts 1) Conductivity depends on E F. 2) E F depends on carrier density. n-type semiconductor 3) So it is common to characterize the conductivity at a given carrier density. 4) obility is often the quantity that is quoted. So we need techniques to measure two quantities: 1) conductivity 2) carrier density 5 2D: conductivity and sheet conductance n-type semiconductor 2D electrons Wt G = σ n L = σ t W n L sheet conductance 6

4 2D electrons vs. 3D electrons n-type semiconductor 3D electrons: 2D electrons: 7 mobility 1) easure the conductivity: 2) easure the sheet carrier density: 3) Deduce the mobility from: 4) Relate the mobility to material parameters: 8

5 recap There are three near-equilibrium transport coefficients: conductivity, Seebeck (and eltier) 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 outline 1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der auw method 5. Summary This work is licensed under a Creative Commons Attribution- oncommercial-sharealike 3.0 United States License. 10

6 2-probe measurements 1 2 R CH = ρ S L W V 21 = I ( 2R C + R CH ) R CH V 21 I 11 transmission line measurements X X X X V ji = I ( 2R C + ρ S S ji W ) H.H. Berger, odels for Contacts to lanar Devices, Solid-State Electron., 15, ,

7 transmission line measurements (TL) Side view i j 13 contact resistance (vertical flow) metal contact Area = A C interfacial layer n-si 14 Side view

8 contact resistance (vertical flow) 10 8 < ρ C < 10 6 Ω-cm 2 interfacial contact resistivity A C = 0.10 µm 1.0µm ρ C = 10 7 Ω-cm 2 R C = 100 Ω n-si Side view interfacial layer 15 contact resistance (vertical + lateral flow) L T transfer length L T = ρ C ρ SD cm (W into page) 16

9 contact resistance 17 transfer length measurments (TL) X Side view X X X 1) Slope gives sheet resistance, intercept gives contact resistance 2) Determine specific contact resistivity and transfer length: 18

10 four probe measurements 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 Hall bar geometry pattern created with photolithography 1 2 thin film isolated from substrate Contacts 0 and 5: current probes Contacts 1 and 2 (3 and 4): voltage probes (high impedance voltmeter) no contact resistance 20

11 outline 1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der auw method 5. Summary This work is licensed under a Creative Commons Attribution- oncommercial-sharealike 3.0 United States License 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 characterize electronic materials. It also finds use magnetic field sensors

12 of a 2D film Hall effect: analysis n-type Hall factor J n = nqµ n E - ( σ n µ n r H ) E B Hall concentration example What are the: 1) resistivity? 2) sheet carrier density? 3) mobility? B = 0: B 0:

13 example: resistivity 3 4 resistivity: B = 0: B = 0.2T: 25 example: sheet carrier density sheet carrier density: B = 0: B = 0.2T: 26

14 example: mobility mobility: B = 0: B = 0.2T: 27 re-cap ) Hall coefficient: 2) Hall concentration: 3) Hall mobility: 4) Hall factor: 28

15 outline 1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der auw method 5. Summary This work is licensed under a Creative Commons Attribution- oncommercial-sharealike 3.0 United States License van der auw sample 2D film arbitrarily shaped homogeneous, isotropic (no holes) Four small contacts along the perimeter 30

16 van der auw approach Resistivity Hall effect B-field = 0 B-field 1) force a current in and out 2) measure V 3) R, = V / I related to ρ S 1) force a current in and out 2) measure V 3) R, = V / I related to V H 31 van der auw approach: Hall effect Hall effect J x = σ n E x - ( σ n µ n r H ) E y B z J y = σ n E y + ( σ n µ n r H ) E x B z E x = ρ n J x + ( ρ n µ H B z ) J y J n = σ n E - ( σ n µ n r H ) E B E y = ( ρ n µ H B z ) J x + ρ n J y V ( B z ) = V H 1 2 V E dl = E x dx +E y dy ( +B z ) V ( B z ) 32

17 van der auw approach: Hall effect Hall effect y V H = ρ n µ H B z J x dy J y dx y x x J n = nqµ n E - ( σ n µ n r H ) E B V H = ρ n µ H B z I I = J i ˆndl So we can do Hall effect measurements on such samples. For the missing steps, see Lundstrom, Fundamentals of Carrier Transport, 2 nd Ed., Sec van der auw approach: resistivity Resistivity semi-infinite half-plane 34

18 van der auw approach: resistivity semi-infinite half-plane 35 van der auw approach: resistivity semi-infinite half-plane but there is also a contribution from contact 36

19 van der auw approach: resistivity semi-infinite half-plane it can be shown that: Given two measurements of resistance, this equation can be solved for the sheet resistance. 37 van der auw approach: resistivity semi-infinite half-plane The same equation applies for an arbitrarily shaped sample! 38

20 van der auw technique: regular sample Force I through two contacts, measure V between the other two contacts. 39 van der auw technique: summary 1) measure n H B-field B-field X 40

21 van der auw technique: summary B = 0 2) measure ρ S 3) determine µ H 41 outline 1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der auw method 5. Summary This work is licensed under a Creative Commons Attribution- oncommercial-sharealike 3.0 United States License. 42

22 summary 1) Hall bar or van der auw geometries allow measurement of both resistivity and Hall concentration from which the Hall mobility can be deduced. 2) Temperature-dependent measurements (to be discussed in the next lecture) provide information about the dominant scattering mechanisms. 3) Care must be taken to exclude thermoelectric effects (also to be discussed in the next lecture). 43 for more about low-field measurments D.K. Schroder, Semiconductor aterial and Device Characterization, 3 rd Ed., IEEE ress, Wiley Interscience, ew York, D.C. Look, Electrical Characterization of GaAs aterials and Devices, John Wiley and Sons, ew York, E. Cage, R.F. Dziuba, and B.F. Field, A Test of the Quantum Hall Effect as a Resistance Standard, IEEE Trans. Instrumentation and easurement, Vol. I-34, pp , 1985 L.J. van der auw, A method of measuring specific resistivity and Hall effect of discs of arbitrary shape, hillips Research Reports, vol. 13, pp. 1-9, Lundstrom, Fundamentals of Carrier Transport, Cambridge Univ. ress, Chapter 4, Sec. 7 44

23 questions 1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der auw method 5. Summary 45