• Rigorous solution to the general problem of calculating sensitivities to local variations – Extended to include local variations in N s , , B as well as R S and R HS . – 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 NN grid as an N 3 process, rather than N 5 – N 2 process for special cases. 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, [email protected], (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
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Rigorous solution to the general problem of calculating sensitivities to local variations –Extended to include local variations in N s, , B as well as.
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• 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 NN grid as an N3 process, rather than N5
– N2 process for special cases.
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, [email protected], (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
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 (single-configuration 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 4PP: A=p2 = square of pitch
of specimen between probes
Sheet resistance:
So, S and f are the same,
Hall sheet resistance: aside from the definition of A.
St. Lawrence University Physics Department, Canton, NY, USA
,
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i
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iij RRA
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,
Analytic form for Resistive and Hall Weighting functions or Sensitivities: linear limit
St. Lawrence University Physics Department, Canton, NY, USA
dA
dAf
A
AR
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Tildes refer to conjugate configuration, i.e. swapping current and voltage leads.
Weighting functions: square vdP & linear 4PP
......
St. Lawrence University Physics Department, Canton, NY, USA
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.
Resistivity
Effect of large inhomogeneity is to use the perturbed local electric field, , instead of the unperturbed value in
This problem can be solved analytically.
For the resistive weighting function,
So, for the extreme case of physical holes in the specimen, nonlinearity is 2 linear effect.
Add nonlinearity of perturbation (zero mag. field)
St. Lawrence University Physics Department, Canton, NY, USA
d
d
dd
dd
GG
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ff
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0/,0/, 1
E
d
T
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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):
Nonzero magnetic induction
St. Lawrence University Physics Department, Canton, NY, USA
.2sin2cos
),~(),~(),(),(
2sin2cos
)~,()~,(),(),(
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0,
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Finite magnetic field, perturbation:
St. Lawrence University Physics Department, Canton, NY, USA
essy.is quite meral case so the gen
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sheetfiniteinFwhere F
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Varying Ns, , B: one at a time
St. Lawrence University Physics Department, Canton, NY, USA
vdPB
Fg
casegeneralBB
FBBgBB
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casegeneralB
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One difference for 4PP vs. vdP
Given that the weighting functions f and g vary with the magnetic field in the small-field limit,
Let’s test this with simulations...
St. Lawrence University Physics Department, Canton, NY, USA
,
2lim
2lim
lim
0,
0,0,
00
1,
0,
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1,
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i
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SNSN
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.
St. Lawrence University Physics Department, Canton, NY, USA
Best Fit Theory
Resistive, fi,0 1.5717 1.5712
Hall, gi,0 1.447 1.472
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).
St. Lawrence University Physics Department, Canton, NY, USA
Best Fit Theory
Resistive, fi,0 1.465 1.45
Hall, gi,0 -2.8428 -2.81
The most extreme nonlinearity is removing conducting material from some part of the specimen: a physical hole.
So, 100% decrease in local Rs
ductance has 200% the impact of a 1% decrease.
Figure: 25mm diameter, 35m thick, 59010 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.
APPLICATION #1: Physical holes
St. Lawrence University Physics Department, Canton, NY, USA
0,21
0, 2/1
,1/
idd
iRG
dd
fGG
fSso
GG
i
d
APPLICATION #1: Physical holes, continued
Single & dual vdP results.
Least squares fit to left-
most three data points is
shown in the plot. There
should be zero degrees of
freedom in the fit.
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
Fit Expected
RS 584 59010 (measured)
fi,0 2.886 2 / ln 2 2.885 (theory)
A
Af
R
RFIT i
S
S
0,2:
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 NN grid as an N3 process, rather than N5