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.
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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.
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3
Landauer conductance and conductivity
“ideal” contacts
cross-sectional
area, A
n-type semiconductor
4
(diffusive)
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For ballistic or quasi-ballistic
transport, replace the mfp with
the “apparent” mfp:
conductivity and mobility
“ideal” contacts
1) Conductivity depends on EF.
cross-sectional
area, A
2) EF depends on carrier density.
n-type semiconductor
3) So it is common to characterize
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
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2D: conductivity and sheet conductance
2D electrons
n-type semiconductor
⎛ Wt ⎞
⎛W ⎞
G = σ n ⎜ ⎟ = σ nt ⎜ ⎟
⎝ L⎠
⎝ L⎠
Top view
“sheet conductance”
6
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2D electrons vs. 3D electrons
Top view
n-type semiconductor
3D electrons:
2D electrons:
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mobility
1) Measure the conductivity:
2) Measure the sheet carrier density:
3) Deduce the mobility from:
4) Relate the mobility to material parameters:
8
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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
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2-probe measurements
Top view
RCH = ρS
1
2
(
V21 = I 2RC + RCH
RCH ≠
11
L
W
)
V21
I
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“transmission line measurements”
Top view
1
2
3
4
5
X
X
X
X
(
V ji = I 2RC + ρS S ji W
)
H.H. Berger, “Models for Contacts to Planar Devices,” Solid-State Electron., 15,
145-158, 1972.
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transmission line measurements (TLM)
1
2
3
4
5
Side view
i
13
j
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contact resistance (vertical flow)
metal contact
Area = AC
interfacial
layer
n-Si
Top view
14
Side view
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contact resistance (vertical flow)
10−8 < ρC < 10−6 Ω-cm 2
“interfacial contact resistivity”
interfacial
layer
AC = 0.10 µ m × 1.0 µ m
ρC = 10−7 Ω-cm 2
n-Si
RC = 100 Ω
Side view
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contact resistance (vertical + lateral flow)
LT
“transfer length” LT =
(W into page)
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ρC
cm
ρSD
contact resistance
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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)
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Hall bar geometry
thin film
isolated from substrate
Top view
pattern created with
photolithography
1
2
5
0
3
4
(high impedance voltmeter)
Contacts 0 and 5: “current probes”
Contacts 1 and 2 (3 and 4): “voltage probes”
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no contact resistance
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 AttributionNonCommercial-ShareAlike 3.0 United States License.
http://creativecommons.org/licenses/by-nc-sa/3.0/us/
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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
22
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ECE-656
F11use magnetic field sensors.
22
Hall effect: analysis
Top view of a 2D film
++++++++++
“Hall factor”
n-type
------------

 
J n = nqµ nE - σ n µ n rH E × B
(
23
)
“Hall concentration”
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example
Top view
1
2
5
0
3
4
What are the:
1) resistivity?
2) sheet carrier density?
3) mobility?
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B = 0:
B ≠ 0:
example: resistivity
Top view
2
1
5
0
3
4
B = 0:
resistivity:
B = 0.2T:
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25
example: sheet carrier density
Top view
1
2
5
0
3
4
B = 0:
sheet carrier density:
B = 0.2T:
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example: mobility
Top view
1
2
5
0
3
4
B = 0:
mobility:
B = 0.2T:
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27
re-cap
Top view
1
1) Hall coefficient:
2
5
0
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 AttributionNonCommercial-ShareAlike 3.0 United States License.
http://creativecommons.org/licenses/by-nc-sa/3.0/us/
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van der Pauw sample
2D film
arbitrarily shaped
homogeneous, isotropic
(no holes)
Top view
P
M
N
O
Four small contacts
along the perimeter
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van der Pauw approach
Resistivity
Hall effect
B-field 
B-field = 0
N
P
M
P
M
O
N
O
1) force a current in M and out N
2) measure VPO
3) RMN, OP = VPO / I related to ρS
1) force a current in M and out O
2) measure VPN
3) RMO, NP = VPN / I related to VH
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van der Pauw approach: Hall effect
Hall effect
(
)
J x = σ nE x - σ n µ n rH E y Bz
(
)
(
)
J y = σ nE y + σ n µ n rH Ex Bz
P
M
E x = ρn J x + ρn µ H Bz J y
O
N
(
)
E y = − ρn µ H Bz J x + ρn J y
P 
P

VPN Bz = − ∫E • dl = − ∫E x dx + E y dy
( )


 
J n = σ nE - σ n µ n rH E × B
(
)
N
N
1
VH ≡ ⎡⎣VPN + Bz − VPN − Bz ⎤⎦
2
(
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)
(
)
32
van der Pauw approach: Hall effect
Hall effect
xP
⎡ yP
⎤
VH = ρn µ H Bz ⎢ ∫ J x dy − ∫ J y dx ⎥
⎢⎣ yN
⎥⎦
xN
VH = ρn µ H Bz I
O
N

I = ∫ J i n̂ dl
P
P
M
N
So we can do Hall effect
measurements on such samples.


 
J n = nqµ nE - σ n µ n rH E × B
(
)
For the missing steps, see Lundstrom,
Fundamentals of Carrier Transport, 2nd Ed.,
Sec. 4.7.1.
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van der Pauw approach: resistivity
Resistivity
semi-infinite half-plane
P
M
N
O
M
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N
O
P
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van der Pauw approach: resistivity
semi-infinite half-plane
M
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van der Pauw approach: resistivity
semi-infinite half-plane
M
N
O
P
but there is also a
contribution from contact N
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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.
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van der Pauw approach: resistivity
semi-infinite
half-plane
M
N
O
P
M
P
N
O
The same equation applies for an arbitrarily shaped sample!
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van der Pauw technique: regular sample
Top view
M
P
N
O
Force I through two contacts, measure V between the other two contacts.
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van der Pauw technique: summary
M
P
M
P
N
O
1) measure nH
N
O
B-field 
40
B-field X
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van der Pauw technique: summary
B=0
M
P
M
P
N
O
2) measure ρS
N
41
O
3) determine µH
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 AttributionNonCommercial-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).
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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
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44
questions
1. Introduction
2. Resistivity / conductivity measurements
3. Hall effect measurements
4. The van der Pauw method
5. Summary
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