l17 resistivity and hall effect measurments - nanohubl17...resistivity and hall effect measurements...
TRANSCRIPT
ECE-656: Fall 2011
Lecture 17:
Resistivity and Hall effect Measurements
Mark Lundstrom Purdue University
West Lafayette, IN USA
1 10/5/11
2
measurement of conductivity / resistivity
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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.
3
resistivity / conductivity measurements
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.
For uniform carrier concentrations:
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diffusive transport assumed
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Landauer conductance and conductivity
n-type semiconductor
cross-sectional area, A
“ideal” contacts
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For ballistic or quasi-ballistic transport, replace the mfp with the “apparent” mfp:
(diffusive)
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conductivity and mobility
n-type semiconductor
cross-sectional area, A
“ideal” contacts
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1) Conductivity depends on EF.
2) EF depends on carrier density.
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
2D electrons
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2D: conductivity and sheet conductance
n-type semiconductor
G = σ n
WtL
⎛⎝⎜
⎞⎠⎟= σ nt
WL
⎛⎝⎜
⎞⎠⎟
“sheet conductance”
Top view
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2D electrons vs. 3D electrons
n-type semiconductor
Top view
3D electrons:
2D electrons:
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mobility
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1) Measure the conductivity:
2) Measure the sheet carrier density:
3) Deduce the mobility from:
4) Relate the mobility to material parameters:
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recap
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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.
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outline
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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/
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
V21 = I 2RC + RCH( )
Top view
2 1 RCH = ρS
LW
RCH ≠
V21
I
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“transmission line measurements”
Top view
2 1 3 4 5
H.H. Berger, “Models for Contacts to Planar Devices,” Solid-State Electron., 15, 145-158, 1972.
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X X
X X
Vji = I 2RC + ρS S ji W( )
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transmission line measurements (TLM)
Side view
2 1
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3 4 5
j i
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contact resistance (vertical flow)
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metal contact
Area = AC
n-Si
Top view Side view
interfacial layer
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contact resistance (vertical flow)
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n-Si
Side view
interfacial layer
10−8 < ρC < 10−6 Ω-cm2
“interfacial contact resistivity”
AC = 0.10 µm ×1.0µm
ρC = 10−7 Ω-cm2
RC = 100 Ω
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contact resistance (vertical + lateral flow)
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(W into page)
LT “transfer length” LT =
ρC
ρSD
cm
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contact resistance
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transfer length measurments (TLM)
Side view
2 1 3 4 5
X X
X X
1) Slope gives sheet resistance, intercept gives contact resistance
2) Determine specific contact resistivity and transfer length:
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four probe measurements
Side view
1 2 3 4
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
Top view
1 2
3 4
0 5
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Contacts 0 and 5: “current probes” Contacts 1 and 2 (3 and 4): “voltage probes”
thin film isolated from substrate
pattern created with photolithography
(high impedance voltmeter)
no contact resistance
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outline
Lundstrom ECE-656 F11
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/
1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der Pauw method 5. Summary
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Hall effect
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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.
n-type semiconductor
current in x-direction:
B-field in z-direction:
Hall voltage measured in the y-direction:
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Hall effect: analysis
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n-type - - - - - - - - - - - - -
+ + + + + + + + + +
Jn = nqµn
E - σ nµnrH( ) E ×
B
Top view of a 2D film
“Hall concentration”
“Hall factor”
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example
Top view
1 2
3 4
0 5
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B = 0:
B ≠ 0:
What are the: 1) resistivity? 2) sheet carrier density? 3) mobility?
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example: resistivity
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Top view 1 2
3 4
0 5
B = 0:
B = 0.2T:
resistivity:
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example: sheet carrier density
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Top view 1 2
3 4
0 5
B = 0:
B = 0.2T: sheet carrier density:
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example: mobility
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Top view 1 2
3 4
0 5
B = 0:
B = 0.2T:
mobility:
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re-cap
Top view 1 2
3 4
0 5
1) Hall coefficient:
2) Hall concentration:
3) Hall mobility:
4) Hall factor:
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outline
Lundstrom ECE-656 F11
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/
1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der Pauw method 5. Summary
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van der Pauw sample
M
N O
P
Top view 2D film arbitrarily shaped homogeneous, isotropic (no holes)
Four small contacts along the perimeter
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van der Pauw approach
M
N O
P
Resistivity
1) force a current in M and out N 2) measure VPO 3) RMN, OP = VPO / I related to ρS
B-field = 0
Hall effect
M
N O
P
1) force a current in M and out O 2) measure VPN 3) RMO, NP = VPN / I related to VH
B-field
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van der Pauw approach: Hall effect Hall effect
M
N O
P
Jn = σ n
E - σ nµnrH( ) E ×
B
Jx = σ nE x - σ nµnrH( )Ey Bz
J y = σ nE y + σ nµnrH( )Ex Bz
E x = ρn Jx + ρnµH Bz( ) J y
E y = − ρnµH Bz( ) Jx + ρn J y
VPN Bz( ) = −
E •
N
P
∫ dl = − E x dx +E y dy
N
P
∫
VH ≡
12
VPN +Bz( ) −VPN −Bz( )⎡⎣ ⎤⎦
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van der Pauw approach: Hall effect
Hall effect
M
N O
P
Jn = nqµn
E - σ nµnrH( ) E ×
B
VH = ρnµH Bz Jx dy − J y dx
xN
xP
∫yN
yP
∫⎡
⎣⎢⎢
⎤
⎦⎥⎥
I =
J in dl
N
P
∫ VH = ρnµH Bz I
So we can do Hall effect measurements on such samples.
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
M
N O
P
Resistivity
M N O P
semi-infinite half-plane
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van der Pauw approach: resistivity
semi-infinite half-plane
M
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van der Pauw approach: resistivity
M N O P
semi-infinite half-plane
but there is also a contribution from contact N
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van der Pauw approach: resistivity
M N O P
semi-infinite half-plane
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
M N O P
semi-infinite half-plane
The same equation applies for an arbitrarily shaped sample!
M
N O
P
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van der Pauw technique: regular sample
Top view
Force I through two contacts, measure V between the other two contacts.
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N O
P M
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P
ON
M
van der Pauw technique: summary
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B-field
P
O N
M
B-field X
1) measure nH
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P
ON
M
van der Pauw technique: summary
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B = 0
P
O N
M
2) measure ρS
3) determine µH
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outline
Lundstrom ECE-656 F11
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/
1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der Pauw method 5. Summary
summary
43 Lundstrom ECE-656 F11
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).
for more about low-field measurments
D.C. Look, Electrical Characterization of GaAs Materials and Devices, John Wiley and Sons, New York, 1989.
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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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D.K. Schroder, Semiconductor Material and Device Characterization, 3rd Ed., IEEE Press, Wiley Interscience, New York, 2006.
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
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questions
1. Introduction 2. Resistivity / conductivity measurements 3. Hall effect measurements 4. The van der Pauw method 5. Summary