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 V = I 2R + R 21 ( C CH )
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 = nqµ E - σ µ r E × B n n ( n n 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 - r E B V ⎡V B V B ⎤ n = σ nE (σ nµn 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 nd n n ( n n H ) Fundamentals of Carrier Transport, 2 Ed., 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.
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).
Lundstrom ECE-656 F11 43
for more about low-field measurments
D.K. Schroder, Semiconductor Material and Device Characterization, 3rd Ed., IEEE Press, Wiley Interscience, New York, 2006. D.C. Look, Electrical Characterization of GaAs Materials and Devices, John Wiley and Sons, New York, 1989. M.E. Cage, R.F. Dziuba, and B.F. Field, “A Test of the Quantum Hall Effect as a Resistance Standard,” IEEE Trans. Instrumentation and Measurement,” Vol. IM-34, pp. 301-303, 1985
L.J. van der Pauw, “A method of measuring specific resistivity and Hall effect of discs of arbitrary shape,” Phillips Research Reports, vol. 13, pp. 1-9, 1958.
Lundstrom, Fundamentals of Carrier Transport, Cambridge Univ. Press, 2000. Chapter 4, Sec. 7
Lundstrom ECE-656 F11 44 questions
1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der Pauw method 5. Summary
Lundstrom ECE-656 F11 45