l17 resistivity and hall effect measurments - nanohubl17...resistivity and hall effect measurements...

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

Lundstrom ECE-656 F11

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:


diffusive transport assumed

4

Landauer conductance and conductivity

n-type semiconductor

cross-sectional area, A

“ideal” contacts


For ballistic or quasi-ballistic transport, replace the mfp with the “apparent” mfp:

(diffusive)

5

conductivity and mobility


cross-sectional area, A

“ideal” contacts


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

6

2D: conductivity and sheet conductance


G = σ n

WtL

⎛⎝⎜

⎞⎠⎟= σ nt

WL

⎛⎝⎜

⎞⎠⎟

“sheet conductance”

Top view


7

2D electrons vs. 3D electrons


Top view

3D electrons:

2D electrons:


8

mobility


1) Measure the conductivity:

2) Measure the sheet carrier density:

3) Deduce the mobility from:

4) Relate the mobility to material parameters:

9

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.

10

outline


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

11

2-probe measurements

V21 = I 2RC + RCH( )

Top view

2 1 RCH = ρS

LW

RCH ≠

V21

I


12

“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.


X X

X X

Vji = I 2RC + ρS S ji W( )

13

transmission line measurements (TLM)

Side view

2 1


3 4 5

j i

14

contact resistance (vertical flow)


metal contact

Area = AC

n-Si

Top view Side view

interfacial layer

15

contact resistance (vertical flow)


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

16

contact resistance (vertical + lateral flow)


(W into page)

LT “transfer length” LT =

ρC

ρSD

cm

17

contact resistance


18

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:

19

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)


20

Hall bar geometry

Top view

1 2

3 4

0 5


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

21

outline




Lundstrom ECE-656 F11 22

Hall effect

22

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.


current in x-direction:

B-field in z-direction:

Hall voltage measured in the y-direction:


Hall effect: analysis

23

n-type - - - - - - - - - - - - -

+ + + + + + + + + +

Jn = nqµn

E - σ nµnrH( ) E ×

B

Top view of a 2D film

“Hall concentration”

“Hall factor”

24

example

Top view

1 2

3 4

0 5


B = 0:

B ≠ 0:

What are the: 1) resistivity? 2) sheet carrier density? 3) mobility?

25

example: resistivity


Top view 1 2

3 4

0 5

B = 0:

B = 0.2T:

resistivity:

26

example: sheet carrier density


Top view 1 2

3 4

0 5

B = 0:

B = 0.2T: sheet carrier density:

27

example: mobility


Top view 1 2

3 4

0 5

B = 0:

B = 0.2T:

mobility:

28

re-cap

Top view 1 2

3 4

0 5

1) Hall coefficient:

2) Hall concentration:

3) Hall mobility:

4) Hall factor:

29

outline





van der Pauw sample

M

N O

P

Top view 2D film arbitrarily shaped homogeneous, isotropic (no holes)

Four small contacts along the perimeter


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


van der Pauw approach: Hall effect Hall effect

M

N O

P

Jn = σ n


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( )⎡⎣ ⎤⎦


van der Pauw approach: Hall effect

Hall effect

M

N O

P

Jn = nqµn


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.


van der Pauw approach: resistivity

M

N O

P

Resistivity

M N O P

semi-infinite half-plane




M



M N O P


but there is also a contribution from contact N



M N O P


it can be shown that:

Given two measurements of resistance, this equation can be solved for the sheet resistance.



M N O P


The same equation applies for an arbitrarily shaped sample!

M

N O

P

39

van der Pauw technique: regular sample

Top view

Force I through two contacts, measure V between the other two contacts.


N O

P M

40

P

ON

M

van der Pauw technique: summary


B-field

P

O N

M

B-field X

1) measure nH

41

P

ON

M

van der Pauw technique: summary


B = 0

P

O N

M

2) measure ρS

3) determine µH

42

outline




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.

44

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


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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l17 resistivity and hall effect measurments - nanohubl17...resistivity and hall effect measurements...

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