Sensitivity of charge transport measurements
to local inhomogeneities
Daniel W. Koon(a), Fei Wang(b), Dirch Hjorth Petersen(b) , Ole Hansen(b+c)
(a) Physics Dept., St. Lawrence University, dkoon@stlawu.edu,
(b) Department of Micro- and Nanotechnology, Technical University of Denmark, DTU Nanotech,
(c) Danish National Research Foundation’s Center for Individual Nanoparticle Functionality (CINF), DTU
•
•
•
Rigorous solution to the general problem of calculating sensitivities to local
variations
– Extended to include local variations in Ns, , B as well as RS and RHS.
– Confirmed by simulation for both 4PP and vdP
Nonlinear [large] perturbations calculated for a variety of quantities, in zero
and finite fields
– Confirms experimental evidence on physical holes
These equations allow for calculation of sensitivities for arbitrary specimen
geometry modeled on NN grid as an N3 process, rather than N5
– N2 process for special cases.
4-wire resistivity and Hall measurement
•
•
•
One measures charge transport quantities (resistivity, , and Hall
coefficient, RH) by measuring 4-wire resistances
One converts resistances into 2D charge transport quantities (sheet
resistance, RS, and Hall sheet resistance, RHS) by multiplying
resistances by dimensionless geometrical factors, i (singleconfiguration techniques) or by averaging two independent
configurations (dual configuration).
Geometrical factors well-known unless material is of nonuniform
composition. How sensitive is measurement to inhomogeneities?
St. Lawrence University Physics Department, Canton, NY, USA
van der Pauw & four-point probe
van der Pauw [vdP]: (SLU)
Specimen of finite area,
electrodes located at periphery.
Define Resistive, Hall weighting
functions as weights by which
local values are averaged by
measurement.
Advantage: 2 simple functions.
four-point probe [4PP]: (DTU)
Specimen may be finite or
approach limit of infinite size, with
electrodes placed within borders.
Define sensitivity of resistive
measurement to local sheet
resistance, local mobility, carrier
concentration, etc.
Advantage: more rigorous
formalism, more flexible notation.
St. Lawrence University Physics Department, Canton, NY, USA
Weighting functions [vdP], sensitivities [4PP]
Define normalization area for each:
vdP: A= = finite area
of specimen
Sheet resistance:
fi 
4PP: A=p2 = square of pitch
between probes
S R Si , L   i p
R
 Ri / Ri
(  A / A )(  R S , L / R S )
  i
 Ri
 A  R S ,L

p
 A  R S ,L / R S
2

fi.
,
Hall sheet resistance:
gj 
S
Ri
R S ,L
 Ri / Ri
2
So, S and f are the same,
aside from the definition of A.
 Ri / Ri
 A  R HS , L / R HS
St. Lawrence University Physics Department, Canton, NY, USA
Analytic form for Resistive and Hall
Weighting functions or Sensitivities: linear limit
 Ri
2
f i (r )   i A
R S ,L A
  i A  G ( r , r )   G ( r , r )   G ( r , ~
r )   G ( r , ~
r ) 
~
~
J S ,i  J S ,i
Ei Ei
f i (r )  A
 A
~
~
J S ,i  J S ,i d  
E i  E id 


 Ri
2
g i (r )  A
 R HS , L  A
 A  G ( r , r  )   G ( r , r )   G ( ~
r , r )   G ( ~
r , r )   e z
~
~
A J S ,i  J S ,i  e z
A Ei  Ei ez
g i (r ) 

~
~

(
J

J
)

e
d

E

E
 S ,i S ,i z
 i i  e z d 






Tildes refer to conjugate configuration, i.e. swapping current and voltage leads.
St. Lawrence University Physics Department, Canton, NY, USA
Weighting functions: square vdP & linear 4PP
Resistivity
Hall
......
•
•
Regions of negative weighting occur in single-configuration measurements
for sheet resistance, though not for Hall measurement.
These can be eliminated by performing dual-configuration measurement.
St. Lawrence University Physics Department, Canton, NY, USA
Add nonlinearity of perturbation
(zero mag. field)
Effect of large inhomogeneity is to use the perturbed local
electric field, E    , instead of the unperturbed value in
T
 G 
2
  

Gd
This problem can be solved analytically.
For the resistive weighting function,
f i ,G d / G d  0 
f i ,G d / G d  0
1
1 Gd
2 Gd
So, for the extreme case of physical holes in the specimen,
nonlinearity is 2 linear effect.
St. Lawrence University Physics Department, Canton, NY, USA
Nonzero magnetic induction
For the general case of a finite specimen with four
electrodes not at its edges, there is no simple expression
for the B-dependence of f and g. In two specific cases,
however, there is a simple form for the B-dependence: an
infinite sheet and a sheet with electrodes at its boundaries
(the van der Pauw geometry):
St. Lawrence University Physics Department, Canton, NY, USA
Finite magnetic field, perturbation:
Fi
G d  0
G h  0
where F
i,B

1
1
 
Gh
2

Gd
Fi , B
1
1 G d
2 Gd
Fi , 0


 cos 2  H Fi , 0  sin 2  H g i , 0
Gh
gi
where
But
G
d
G h  0
G d  0
g i,B

Gd
1
 
Gh
Gd
2

in finite sheet
vdP geometry
1 G h
2 Gd
g i,B 
1
1
4
Fi , B
 
G h
2
Gd
g i,0


 cos 2  H g i , 0  sin 2  H Fi , 0
 G d  N S ,μ , B   G d  R S ,R HS   G d  R S ,Θ H ,
so the gen eral case
is quite m essy.
St. Lawrence University Physics Department, Canton, NY, USA
in finite sheet
vdP geometry .
Varying Ns, , B: one at a time

2
R
B  0
  0
2


S N iS

B  0
N S  0
 

N S  0
  0
vdP
general
case
vdP



2
2
2
2

1   B 1  BB  g i , B  2  B 1  14 BB (1   B ) Fi , B
B
2
2 2
2
2
1


B
1  14   B

A RS 


A Ri 
B
g

Fi , 0
i
,
0
2
1

sin
2

.
H
2
2
2

1  14   B

R
case


SBi
general

  1   2 B 2 Fi , B  2  B g i , B

2
2 2
1  12  
1  B

A RS 


A Ri 
2
 cos  H Fi , 0

1 

1

2 


R

 

Si

1  B  N S

 1 2
Fi , B   B g i , B
NS

2
2
 1   2 B 2 1  12 NN S  12 NN S  B
S
S
A RS 


A Ri 
N
 Fi , 0  2 cos1  N S f i , B
H
S
 cos 
H
2
2
N
N

1  12 N S  12 N S tan  H
S
S



St. Lawrence University Physics Department, Canton, NY, USA
general
vdP
case
One difference for 4PP vs. vdP
Given that the weighting functions f and g vary with the
magnetic field in the small-field limit,
lim
N S
NS




,  B  1
lim
,  B  1
lim
,  B  1
S
Ri
N S B  0
  0
Ri
S
S
B  0
N S  0
Ri
B B  0
N S  0
A R S   Fi , 0   B g i , 0



A R i   Fi , 0   B g i , 0
A R S   Fi , 0  2  B g i , 0


 Fi , 0

A Ri 
in finite plane
vdP
in finite plane
A R S   B  g i , 0  2  BF i , 0 


 B g i,0

A Ri 
Let’s test this with simulations...
St. Lawrence University Physics Department, Canton, NY, USA
vdP
in finite plane
vdP ,
COMSOL simulation vs theory, 4PP linear array
Perturbation located
0.14p from second
electrode in a linear
4PP.
Agreement between fit
and all data to within
0.001 on main plot.
Best Fit
Theory
Resistive, fi,0
1.5717
1.5712
Hall, gi,0
1.447
1.472
St. Lawrence University Physics Department, Canton, NY, USA
Excel simulation vs theory: vdP square
Probe equidistant from
adjacent current and
voltage probes, 0.3a
from edge of square of
side a.
Decent agreement with
theory, but disastrous fit
to 4PP predictions.
Hall angle, B. same as
for Comsol simulation
(last slide).
Best Fit
Theory
Resistive, fi,0
1.465
1.45
Hall, gi,0
-2.8428
-2.81
St. Lawrence University Physics Department, Canton, NY, USA
APPLICATION #1: Physical holes
The most extreme nonlinearity is removing conducting
material from some part of the specimen: a physical hole.
 G d / G d   1,
so
ductance
S
Ri
Gd

f i,0
1
1
2
Gd / Gd
 2 f i,0
So, 100% decrease in local Rs
has 200% the impact of a 1%
decrease.
Figure: 25mm diameter, 35m thick, 59010 copper foil vdP
specimens with physical holes, from Josef Náhlík, Irena Kašpárková
and Přemysl Fitl, Measurement, Volume 44, Issue 10, December
2011, Pages 1968–1979.
St. Lawrence University Physics Department, Canton, NY, USA
APPLICATION #1: Physical holes, continued
Single & dual vdP results.
Least squares fit to leftmost three data points is
shown in the plot. There
should be zero degrees of
freedom in the fit.
FIT :
RS
RS
 2 f i,0
A
A
Fit
Expected
RS
584
59010 (measured)
fi,0
2.886
2 / ln 2  2.885 (theory)
Surprisingly good fit up to about A/A = 0.25, a hole half
the diameter of the entire specimen, where disagreement
between above fit & exact solution (solid line) is about 9%.
Experimental data: Josef Náhlík, Irena Kašpárková and Přemysl Fitl, Measurement,
Volume 44, Issue 10, December 2011, Pages 1968–1979.
St. Lawrence University Physics Department, Canton, NY, USA
APPLICATION #2:
ZnO charge carrier polarity
• ZnO samples have highly inhomogeneous RS.
• Internal holes in the specimen or radial
inhomogeneities, if electrodes not
located at the edges.
• Can this produce RH of the wrong sign,
thus fool the measurer into imputing
charge carriers of the wrong polarity?
Image: Scanning electron microscopic image of interfacially grown ZnO film.
http://www.chemistry.manchester.ac.uk/groups/pob/research.html. Accessed 2/15/2012.
Citations: Takeshi Ohgaki, N. Ohashi, S. Sugimura, H. Ryoken, I. Sakaguchi, Y.
Adachi, and H. Haneda, “Positive Hall coefficients obtained from contact
misplacement on evident n-type ZnO films and crystals”, J. Mater. Res., 23 (9),
2293-2295 (2008).
Oliver Bierwagen, T. Ive, C. G. Van de Walle, J. S. Speck, “Causes of incorrect
carrier-type identification in van der Pauw-Hall measurements”, App. Phys. Lett.
93, 242108 (2008).
St. Lawrence University Physics Department, Canton, NY, USA
ZnO: Hall effect near interior hole:
electrodes at edge, away from edge
Left: Electrodes at edges
 No regions of negative weighting
 Measured Hall signal lies within range of values within specimen.
Right: Interior electrodes
 Regions of negative weighting
 All bets are off, wrong polarity for Hall signal possible.
Takeshi Ohgaki, N. Ohashi, S. Sugimura, H. Ryoken, I. Sakaguchi, Y. Adachi, and H. Haneda, “Positive Hall
coefficients obtained from contact misplacement on evident n-type ZnO films and crystals”, J. Mater. Res.,
23 (9), 2293-2295 (2008).
St. Lawrence University Physics Department, Canton, NY, USA
ZnO: Hall effect errors for electrodes away from edges
Electrodes in a square array 1/5 the size of the specimen.
Left: Homogeneous specimen. Integral of g5 in negative weighting
regions is 70% the magnitude of integral in positive weighting regions.
Right: Radial inhomogeneities (Carrier density increase 100x from
center to corners in this example.) change the negative contribution to
99% of the positive. Odds of measuring a Hall signal lying outside
values within the specimen rise.
Specimens described in: Oliver Bierwagen, T. Ive, C. G. Van de Walle, J. S. Speck, “Causes of
incorrect carrier-type identification in van der Pauw-Hall measurements”, App. Phys. Lett. 93, 242108
(2008).
St. Lawrence University Physics Department, Canton, NY, USA
Conclusions
• Rigorous solution to the general problem of calculating
sensitivities to local variations
– Extended to include local variations in Ns, , & B as well as RS &
RHS.
– Confirmed by simulation for both 4PP and vdP
• Nonlinear [large] perturbations calculated for a variety of
quantities, in zero and finite fields
– Confirms experimental evidence on physical holes
• These equations allow for calculation of sensitivities for
arbitrary specimen geometry modeled on NN grid as an
N3 process, rather than N5
– N2 process for special cases.
Contact information: dkoon@stlawu.edu