-
Capacitance-voltage profiling: Research-grade approach versus
low-cost alternativesNeal D. Reynolds, Cristian D. Panda, and John
M. Essick Citation: American Journal of Physics 82, 196 (2014);
doi: 10.1119/1.4864162 View online:
http://dx.doi.org/10.1119/1.4864162 View Table of Contents:
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Published by the American Association of Physics Teachers Articles
you may be interested in Capacitance-voltage characteristics of
organic Schottky diode with and without deep traps Appl. Phys.
Lett. 99, 023301 (2011); 10.1063/1.3607955 The origin of anomalous
peak and negative capacitance in the forward bias
capacitance-voltage characteristicsof Au/PVA/n-Si structures J.
Appl. Phys. 109, 074503 (2011); 10.1063/1.3554479
Capacitance–voltage studies of Al-Schottky contacts on
hydrogen-terminated diamond Appl. Phys. Lett. 81, 637 (2002);
10.1063/1.1496495 Barrier height determination of SiC Schottky
diodes by capacitance and current–voltage measurements J. Appl.
Phys. 91, 9841 (2002); 10.1063/1.1477256 Capacitance–voltage and
admittance spectroscopy of self-assembled Ge islands in Si Appl.
Phys. Lett. 77, 2704 (2000); 10.1063/1.1320036
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Capacitance-voltage profiling: Research-grade approachversus
low-cost alternatives
Neal D. Reynolds, Cristian D. Panda, and John M. Essicka)
Physics Department, Reed College, Portland, Oregon 97202
(Received 21 July 2013; accepted 22 January 2014)
We describe an experiment that implements capacitance-voltage
profiling on a reverse-biased
Schottky barrier diode to determine the density of impurity
dopants in its semiconductor layer as well
as its built-in electric potential. Our sample is a commercially
produced Schottky diode. Three
different experimental setups, one using research-grade
instrumentation, the other two using low-cost
alternatives, are given and their results compared. In each of
the low-cost setups, phase-sensitive
detection required to measure the sample’s capacitance is
carried out using an inexpensive data
acquisition (DAQ) device and a software program that implements
a lock-in detection algorithm. The
limitations of the DAQ device being used (e.g., restricted
analog-to-digital conversion speed,
inadequate waveform generation capabilities, lack of hardware
triggering) are taken into account in
each setup. Excellent agreement for the value of the doping
density obtained by the all three setups is
found and this value is shown to be consistent with the result
of an independent method (secondary
ion mass spectroscopy). VC 2014 American Association of Physics
Teachers.
[http://dx.doi.org/10.1119/1.4864162]
I. INTRODUCTION
Semiconductor physics has proved itself a fertile field
foradvanced instructional laboratory developers. The properties
ofsemiconductors engender natural student interest as they lie
atthe heart of modern-day technologies such as computers,
smartphones, and the internet, while, pedagogically,
semiconductorphenomena provide engaging applications of basic
concepts inquantum mechanics, electrodynamics, and thermal
physics.Hence, many semiconductor-related advanced
laboratoryexperiments have been developed. For example,
electricalmeasurements on forward-biased diodes and Hall devices
havebeen used to measure the semiconducting band gap,1–6
Schottky barrier height,7 charge carrier density,8–10
carriertransport properties,11,12 and carrier statistical
distributions.13
Additionally, optical experiments are available to determine
theband gap of bulk,14,15 thin film,16 and quantum-dot17
semicon-ductor samples, the Maxwell-Boltzmann distribution of
chargecarriers,18 and the vibration properties of
nanomaterials.19
In this paper, we describe a newly developed instructionallab
experiment that implements capacitance-voltage (CV)profiling on a
reverse-biased Schottky barrier diode to deter-mine the density of
impurity dopants in its semiconductorlayer, as well as its built-in
electric potential. In contrast tothe more complicated
dual-semiconductor structure of a pn-junction diode, a Schottky
barrier diode consists of a singlesemiconductor layer in contact
with a metallic layer. Due tothe simplicity of its construction,
the functioning of aSchottky barrier diode can be explained
theoretically usingonly introductory electrodynamics and
semiconductor con-cepts. As with all semiconductor devices,
fabrication of ahigh-quality Schottky diode requires specialized
expertiseand expensive deposition systems. To avoid this hurdle,
weuse a commercially produced Schottky barrier diode in
thisproject. Additionally, in an effort to contain the cost of
thecapacitance characterization system, we describe severaloptions
for the setup, ranging from one consisting solely ofstand-alone
research-grade equipment to others that uselow-cost op-amp
circuitry and affordable computer-basedinstrumentation.
II. THEORY
A. Capacitance-voltage profiling of Schottky barrier
In this section, we derive the theoretical relations that arethe
basis of the capacitance-voltage profiling
experimentaltechnique.20,22,23,25 First, consider a Schottky
barrier ofcross-sectional area A that consists of a metal layer in
contactwith an n-type semiconductor (dielectric constant �; for
sili-con �¼ 11.7) and define an x-axis whose origin is at
themetal-semiconductor interface with its positive direction
to-ward the semiconductor’s interior (Fig. 1). For the
Fig. 1. Band bending in a Schottky barrier (cross-sectional area
A) underbias of –V0 creates a (shaded) depletion region of width W
with space chargedensity þeq due to ionized dopant atoms; q may be
position-dependent.When the reverse bias is increased by dVR, an
additional charge ofdQ¼þeq(AdW) is created at the tail of the
depletion region. Themetal-semiconductor interface is at x¼ 0 and
it is assumed each ionizeddopant atom has charge of þe.
196 Am. J. Phys. 82 (3), March 2014 http://aapt.org/ajp VC 2014
American Association of Physics Teachers 196
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semiconducting material, we assume the following.
First,positively charged dopant atoms are incorporated in its
lat-tice structure with a position-dependent volume number den-sity
(“doping density”) q(x). Second, its temperature is highenough so
that these dopant atoms are fully ionized, that is,their extra
electrons have all been promoted into the semi-conductor’s
conduction band. These negative conductionelectrons then perfectly
compensate the dopant atom’s posi-tive charge so that, in its bulk,
the semiconductor is electri-cally neutral. Third, at large x the
semiconductor isconnected to external circuitry via an ohmic “back
contact.”
We further assume that the type of metal is properly cho-sen so
that when the two materials are joined, semiconductorconduction
electrons transfer to the metal’s surface, leavingbehind a
positively charged layer of uncompensated dopantatoms (called the
“depletion region”) in the volume of thesemiconductor nearest the
metal. This charge transfer takesplace until the resulting space
charge produces an electricpotential –Vbi at x¼ 0 (Vbi� 0 is termed
the “built-inpotential”), which prevents further flow of charge
(becausethe Fermi levels of the two materials have been
equalized).
Then, if an externally applied voltage –VR (where VR� 0is called
the “reverse bias”) is applied at the metal contact sothat the
total potential at the metal-semiconductor interfaceis –V0¼ –
(VRþVbi) and the total charge on the metal’s sur-face is –Q,
movement of semiconductor conduction electronsaway from the
interface (and out the back contact at large x)will extend the
depletion region to a width W. Beyond thedepletion region, the
semiconductor is neutral and the effectof –Q is not felt (i.e., the
electric field is zero there). Thiseffect is called dielectric
screening and W is the “screeninglength.”
Let’s now analyze the special case of uniform doping
densityq(x)¼ q0 (a constant), where each dopant atom is assumed
tohave a single charge þe. If the cross-sectional dimensions of
theSchottky barrier are much greater than W, then by symmetry wecan
assume the electric field E in the depletion region points inthe
(negative) x-direction. Using the rectangular Gaussian sur-face
shown in Fig. 2, with one of its end caps in the neutral bulkregion
(where E¼ 0) and the other end cap located a distance xfrom the
metal-semiconductor interface, Gauss’s law gives�EA ¼ þeq0AðW �
xÞ=�0. For the electric potential V(x) as afunction of distance x
into the depletion region, we have the two
boundary conditions: Vð0Þ ¼ �ðVR þ VbiÞ and V(W)¼ 0. Thus,from
VðWÞ � Vð0Þ ¼ �
ÐW0
E � dx, we find
VR þ Vbi ¼eq02��0
W2 (1)
and so the width of the depletion region required to screenout
the reverse bias VR is
W ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2��0ðVR
þ VbiÞ
eq0
s: (2)
If the reverse bias VR is increased by a small amount
dVR,(remembering Vbi is a constant) we find from Eq. (1) thatdVR ¼
ðeq0=��0ÞW dW, where dW is the increase in thedepletion region’s
width. This increase in the depletionregion width is due to flow of
conduction electrons at theedge of the depletion region into the
semiconductor’s bulk(and out of the back contact), creating the
extra space chargedQ ¼ þeq0ðA dWÞ required to screen out the
voltage change–dVR at the metal-semiconductor interface. By
definition,this process produces a capacitive response given by
C � dQdVR¼ eq0ðAdWÞðeq0=��0ÞW dW
¼ ��0AW
; (3)
or, using Eq. (2), the Schottky barrier’s capacitance as
afunction of reverse bias VR is
C ¼
Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e��0q02ðVR þ VbiÞ
r: (4)
This expression can be re-written as
1
C2¼ 2
A2e��0q0ðVR þ VbiÞ; (5)
suggesting the following capacitance-voltage characteriza-tion
method: For a Schottky barrier with uniform dopingdensity, a plot
of 1/C2 versus VR will yield a straight linewith slope m ¼
2=A2e��0q0 and y-interceptb ¼ 2Vbi=A2e��0q0. Then, the doping
density and built-inpotential are found from
q0 ¼2
A2e��0m; (6)
and
Vbi ¼b
m: (7)
The general case of position-dependent doping densityq(x) can be
solved as follows: We start with the identity
d
dxx
dV
dx
� �¼ dV
dxþ x d
2V
dx2; (8)
and note from Poisson’s equation that
d2V
dx2¼ þeqðxÞ
��0: (9)
Substituting Eq. (9) into Eq. (8) and integrating both sides
ofthe resulting expression from x¼ 0 to x¼W then yields
VR þ Vbi ¼e
��0
ðW0
xqðxÞdx: (10)
Fig. 2. Gauss’s Law applied to a Schottky barrier
(cross-sectional area A) underbias of –V0¼ –(VRþVbi) to determine
the electric field E at location x, assum-ing constant doping
density. By symmetry, the electric field E is directed alongthe
(negative) x-axis. A (dashed) rectangular Gaussian surface is
chosen withone face in the depletion region, where the space charge
density is þeq0; theother face is in neutral bulk region of the
semiconductor, where the electric field
is zero. The total charge within the Gaussian surface is dQ¼þeq0
A(W – x).
197 Am. J. Phys., Vol. 82, No. 3, March 2014 Reynolds, Panda,
and Essick 197
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Now if the reverse bias VR is increased by a small amountdVR,
the depletion region width will change by dW, creatingthe extra
space charge dQ ¼ þeqðWÞðA dWÞ, where q(W) isthe doping density at
the edge of the depletion region whenthe reverse bias is VR. From
Eq. (10), we have
dðVR þ VbiÞ ¼e
��0d
ðW0
xqðxÞdx !
; (11)
or
dVR ¼e
��0WqðWÞ dW: (12)
Thus, the capacitive response is given by
C � dQdVR¼ þe qðWÞðA dWÞðe=��0ÞW qðWÞ dW
¼ ��0AW
: (13)
Re-writing this expression as 1=C2 ¼ ðW=��0AÞ2 and
differ-entiating with respect to VR gives
d
dVR
1
C2
� �¼ 1ð��0AÞ2
2WdW
dVR; (14)
or, using Eq. (12),
d
dVR
1
C2
� �¼ 1ð��0AÞ2
2W��0
eW qðWÞ
� �¼ 2
A2e��0 qðWÞ:
(15)
Equation (15) is called the “Profiler’s Equation” and canbe used
to characterize the spatial distribution of dopants inthe
semiconductor as follows: Starting with data for theSchottky
barrier’s capacitance C as a function of reverse biasVR, a plot of
1/C
2 versus VR is constructed. At each value ofreverse bias VR on
this plot, the slope m ¼ dð1=C2Þ=dVR isdetermined and the
associated value of capacitance C noted.Then, each value of VR
corresponds to probing the dopingdensity at the distance W from the
metal-semiconductorinterface, which is given by Eq. (13) to be
W ¼ ��0AC
: (16)
Using Eq. (15), the doping density at this distance W is
deter-mined by
qðWÞ ¼ 2A2e��0m
: (17)
To carry out capacitance-voltage characterization of aSchottky
barrier diode, a negative dc voltage –VR is applied toits metal
contact with the back contact grounded, producing aspace-charge
region of width W in the semiconductor. Thebarrier’s capacitance C
is then determined by adding a smallac modulation of amplitude Vac
and angular frequency x tothe applied voltage. To account for a
small leakage currentthrough the barrier and the resistance of the
semiconductor’sneutral bulk region, the Schottky diode is modeled
as the par-allel combination of the capacitor C and a leakage
resistor RL,in series with resistance RS (see Fig. 3). In most
cases, thesemiconductor doping density is large enough so that RS
isnegligible in comparison to the parallel combination of C and
RL. Taking RS� 0, ac circuit analysis then predicts that
theamplitude I of the total ac current flowing in this circuit
is
I ¼ VacZ¼ Vac
1
RLþ ixC
� �� Ix þ Iy: (18)
Hence, relative to the applied ac voltage, the current willhave
an in-phase component Ix proportional to 1/RL and a90� out-of-phase
(“quadrature”) component Iy proportionalto xC. For a high-quality
diode, the leakage current is small(RL� 1/xC) so that Ix Iy.
B. Lock-in detection algorithm
As we have seen, for the circuit described by Eq.
(18),phase-sensitive detection of current is required to measure
(asignal proportional to) capacitance. In our experimental set-ups,
this phase-sensitive detection will be accomplished byusing a
lock-in amplifier,26,27,32–35 which functions as fol-lows: Assume
that in response to an ac modulation inputvoltage Vac sinðxsigtÞ,
an experimental system produces an“in-phase” (relative to the
input) output signal Vsig sinðxsigtÞ,where Vsig and xsig are the
signal’s amplitude and angularfrequency, respectively. If we
construct an “in-phase refer-ence” sinusoid 2 sinðxref tÞ of
amplitude 2 and angular fre-quency xref, and take the product of
the output signal andreference, we obtain
2Vsig sinðxsigtÞsinðxref tÞ ¼ Vsig½cosðxsig � xrefÞt� cosðxsig þ
xrefÞt; (19)
where we used the identity sin a sin b ¼ 1=2½cosða� bÞ�cosðaþ
bÞ. Thus, the multiplicative result is two ac sinu-soids, each of
amplitude Vsig, one with the “difference” fre-quency ðxsig � xrefÞ
and one with the “sum” frequencyðxsig þ xrefÞ. Note that for the
special case xref¼xsig, thedifference-frequency sinusoid is the dc
voltage Vsig. Hence,if the experiment’s output waveform consists of
a collection
Fig. 3. Equivalent circuit of the CV characterization setup. The
reverse-
biased Schottky barrier diode is modeled as capacitance C in
parallel withleakage resistance RL. The semiconductor’s neutral
bulk region (beyond thedepletion region) contributes a series
resistance RS. For a high-quality diode,the impedance of RL is much
greater than that of C and RS is negligible.Thus, the response of
the diode to the small ac modulation voltage is pre-
dominately due to C.
198 Am. J. Phys., Vol. 82, No. 3, March 2014 Reynolds, Panda,
and Essick 198
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of component sinusoids of various frequencies, by multiply-ing
this waveform by 2sinðxref tÞ and then using low-pass fil-tering to
find only the resultant dc value, one can determinethe amplitude of
the “in-phase” sinusoidal component withinthe output waveform whose
frequency equals that of thereference.
In a similar way, if the experimental system produces
a“quadrature” output signal Vsig cosðxsigtÞ in response to theinput
voltage Vac sinðxsigtÞ, by multiplying this output signalby the
“quadrature reference” 2 cosðxref tÞ and low-pass fil-tering for
the resultant dc value, this dc value will equal Vsig(i.e., the
quadrature signal amplitude).
III. EXPERIMENTAL SETUPS AND RESULTS
Our goal is to demonstrate both research-grade and low-cost(but
accurate) implementations of the capacitance profilingtechnique for
use in instructional laboratories. We first carryout the
capacitance-voltage (CV) method using research-grade,stand-alone
instrumentation in order to show how the techniqueworks, and also
to establish a baseline of precision by which tojudge the accuracy
of our lower-cost setups. Next, we showtwo inexpensive
computer-based versions of this experiment,each based on an
affordable USB-interfaced data acquisitiondevice, which produce
excellent results.
For each variant of the experiment, our sample is a
com-mercially produced Schottky diode. Because the measuredsignal
is proportional to the capacitor’s cross-sectional area A,we chose
a Schottky diode with a large-area metallic contact[Semiconix
Semiconductor STPS20120D (Ref. 28)]. The con-tact is composed of a
TiW alloy. We stripped the epoxyencapsulation from one of these
devices29 to expose the diodeand used a calibrated microscope to
measure the square con-tact30 to have a side length of (2.32 6
0.02) mm, yieldingA¼ 5.38 6 0.09 mm2. On this exposed diode,
secondary ionmass spectroscopy (SIMS)31 established that its n-type
siliconlayer has a phosphorous doping density of approximately3�
1015 dopants/cm3, while no arsenic (another commonn-type dopant
atom in silicon) was detected.
At zero applied bias, the capacitance of the STPS20120Ddiode is
on the order of 1000 pF. For all of the experimentsdescribed in
this paper, the ac modulation amplitude and fre-quency are Vac¼ 30
mV rms and f¼ 1000 Hz, respectively,and the reverse bias VR is
scanned over the range from0.0–9.9 V in increments of 0.1 V. To
allow for settling of themeasured signal, a wait of 10 lock-in time
constants is takenbefore reading each data point during the scan.
The modula-tion amplitude is chosen to be less than the
reverse-bias in-crement so that each value of VR in our scan probes
a uniqueposition in our sample; the choice of frequency is dictated
bythe maximum sampling rate of the USB-6009 DAQ device(see
discussion below).
A. Research-grade implementation
A schematic diagram of our research-grade experimentalsetup is
shown in Fig. 4. A USB-interfaced Agilent 33210 Afunction generator
applies the ac oscillation of small ampli-tude Vac and frequency f
along with a dc offset –VR to theSchottky diode’s metal contact.
The other end of the diode isconnected to a DL Instruments 1211
current preamplifier,which is a virtual electrical ground and
converts the diode’scurrent I to a voltage V¼bI, where the
proportionality con-stant b¼ 106 V/A. For the diode and data-taking
parameters
we use, the resulting value for V is on the order of b�ðVac xCÞ
¼ 106 V=A� ½ð30 mV rmsÞ2pð1000 HzÞð10�9 FÞ
� 200 mV rms. This voltage is then read by aGPIB-interfaced
Stanford Research System SR830 lock-inamplifier. The function
generator’s TTL sync output is usedas the lock-in’s reference
signal, with zero phase defined bythe moment of the negative-going
zero crossing of the ac os-cillation (that is, the moment at which
the reverse bias beginsto increase). Since, at constant frequency,
the lock-in meas-ures a quadrature voltage that is proportional
(via b) to thecircuit’s quadrature current, which in turn is
proportional toC [see Eq. (18)], a calibration capacitor36 of known
capaci-tance C0 is substituted in the circuit for the Schottky
diodeand the resultant quadrature voltage output V0y is
measured.Then, with the diode replaced back into the circuit, its
capac-itance C in response to a particular reverse bias is
determinedby measuring the quadrature voltage Vy and using the
pro-portionality relation: C=C0 ¼ Vy=V0y. Data taking is
con-trolled by a LabVIEW program.
To account for small (on the order of a few degrees in
mostcases) phase shifts due to other sources (e.g., cabling,
ampli-fiers) than the capacitances of interest, a further
refinementcalled “autophasing” can be included as follows: With
thecalibration capacitor replacing the Schottky diode in the
cir-cuit, record Vx0 and Vy0, which are the lock-in’s in-phase
andquadrature readings for the known calibration capacitor
C0,respectively. Then, assuming the calibration capacitor has
apurely capacitive impedance (that is, it has zero leakage
cur-rent), these two readings determine a vector in the
compleximpedance plane that defines the direction of the
sample’scapacitive response. With the diode back in the circuit,
itsin-phase and quadrature voltages Vx and Vy are measured.These
two readings determine a vector describing the sam-ple’s ac
response in the complex impedance plane. By mathe-matically finding
the component of the sample’s responsealong the purely capacitive
direction (via a dot product), therelation for the sample’s
capacitance is obtained as
C ¼ ðVxVx0 þ VyVy0ÞV2x0 þ V2y0
C0: (20)
Fig. 4. Research-grade implementation of capacitance profiling
method
using stand-alone instrumentation (Agilent 33210 A function
generator, DL
Instruments 1211 current preamplifier, Stanford Research System
SR830
lock-in amplifier).
199 Am. J. Phys., Vol. 82, No. 3, March 2014 Reynolds, Panda,
and Essick 199
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Commercial research-grade lock-in amplifiers commonlyoffer
autophasing with the push of a button. For software-based lock-ins,
autophasing can be accomplished by includ-ing Eq. (20) in the
software program.
The room-temperature capacitance versus reverse biasdata
acquired on a STPS20120D diode, along with the result-ing 1/C2 vs
VR plot, are shown in Fig. 5. The straight-linecharacter of the
latter plot indicates that the diode’s dopingdensity is constant
over the spatial region profiled. UsingEqs. (6) and (7), the slope
and y-intercept of this plot deter-mine that the doping density in
this region and the diode’sbuilt-in potential are 2.7� 1015
dopant/cm3 and 0.622 V,respectively (in semiconductor physics, it
is typical to givedoping density in units of dopant/cm3, rather
than dop-ant/m3). Our value for Vbi agrees with published
valuesobtained on titanium-tungsten silicide Schottky
contacts.37
The uncertainties in our q and Vbi values obtained fromEqs. (6)
and (7) were estimated as follows. First, the influ-ence of random
measurement error in introducing uncer-tainty in our straight-line
fit of the 1/C2 versus VR plot wasgauged by comparing the results
of ten identical runs of theexperiment. We found that such random
errors contributeduncertainty in m and b on the order of only 0.1%.
Thus, inusing Eq. (6), the uncertainty in the contact area is the
domi-nant contribution to dq. With dA/A� 0.02, we getdq¼ 0.1� 1015
dopant/cm3. To determine Vbi via Eq. (7),only the highly accurate m
and b values are involved. Withdm=m � db=b � 0:001, our model
predicts dVbi � 0:001 V.However, one might question whether our
model for theSchottky barrier, which ignores secondary effects such
as se-ries resistance and small temperature corrections, describes
areal-life diode to this level of accuracy (literature values
forVbi obtained from CV measurements are typically given withonly
two digits of precision).20,24
Using these same CV data, Eqs. (16) and (17) yield theq(W)
versus W plot shown in Fig. 6. Also shown in this plotare the SIMS
data for phosphorous dopant density versus dis-tance from the
metal-semiconductor interface taken on thissample. The SIMS
detection limit for phosphorous is 1� 1015dopant/cm3, so the SIMS
data taken on our sample are some-what noisy. Averaged over the
spatial region from W¼ 0.5 to1.0 lm, the SIMS doping density is (3
6 2)� 1015 dop-ant/cm3. Thus, the doping densities determined by
the capaci-tance and SIMS characterization techniques are
consistent.
Finally, we observe in Fig. 6 that the capacitance
profilingvalue for q(x) becomes more noisy as W increases.
Thiswell-known effect21,38 is explained as follows. FromEq. (17),
the uncertainty dq in the doping density determinedat each location
x¼W is given by dq/q¼ dm/m, where dm isthe uncertainty in
determining the slope m ¼ dð1=C2Þ=dVRat that location (for this
calculation, the same value for A isused at all locations, so the
uncertainty in contact area doesnot contribute to the observed
scatter of q-values). Writing min terms of our measured quantities,
we see that the slope at
Fig. 5. Experimental data (C vs VR and 1/C2 vs VR) taken at room
tempera-
ture on a STPS20120D Schottky diode using research-grade
instrumentation
(time constant of lock-in is s¼ 300 ms). The latter plot
determines thediode’s constant doping density and built-in
potential to be 2.7� 1015 dop-ants/cm3 and 0.622 V, respectively.
Error bar for each data point is smaller
than symbol used to represent point.
Fig. 6. Spatial profile of phosphorous doping. The capacitance
profiling result (solid circles) is found using Eqs. (16) and (17),
where the x-axis is plotting W.The minimum value of W is determined
by Vbi. The SIMS result (open circles) gives an average of 3� 1015
dopant/cm3 in the region that overlaps with thatprofiled by the
capacitance method.
200 Am. J. Phys., Vol. 82, No. 3, March 2014 Reynolds, Panda,
and Essick 200
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x¼W is calculated by m � ð�2=C3ÞDC=DVR, where DCand DVR are the
differences between neighboring values ofcapacitance and reverse
bias at that location, respectively. Inour data scans, DVR¼ 0.100 V
at all locations and so theuncertainty in this quantity cannot
explain the increase in dqat increasing W. Conversely, as shown in
Fig. 5, the CVcurve flattens out at larger biases and thus the
difference DCin neighboring capacitance values is smaller and more
sus-ceptible to random measurement error as VR (and its associ-ated
W) increases. Keeping just this relevant contribution
touncertainty, we get dm=m � dðDCÞ=DC. Since DC is the dif-ference
between two neighboring capacitance values, eachwith an uncertainty
of dC, we find dðDCÞ ¼
ffiffiffi2p
dC. Also,from Eq. (13), we have DC � ð���0A=W2ÞDW. PuttingEq.
(12) into this expression, we find DC � ð��2�20A=eqW3ÞDVR, and
thus
dqq¼ dm
m� dðDCÞ
DC�
ffiffiffi2p
dCð��2�20A=eqW3ÞDVR
� �ffiffiffi2p
e
�2�20A
!qdCDVR
� �W3: (21)
In our scans, dC and DVR are constant and q is found to
beuniform in our sample. Thus, Eq. (21) predicts that dq will
beproportional W3, explaining why the doping density
becomesmarkedly more noisy as we profile deeper into the
sample(alternate profiling methods have been developed that
useconstant electric field increments, rather than constant
reversebias steps, which somewhat alleviate this noise problem
atlarger profiling depths). From the data in Fig. 6 we
determinethat in the range W¼ 2.00–2.25 lm, dq¼ 4.4� 1013
dop-ant/cm3. Using this value, along with the other known
quanti-ties in Eq. (21), we find that dC¼ 0.02 pF is responsible
forthe observed scatter in q.
B. Low-cost implementation
We now demonstrate that similar results can be obtainedfrom two
different low-cost experimental setups. In each ofthese setups, the
commercial current preamplifier is replaced
by a simple current-to-voltage (I-to-V) op-amp circuit with a106
X feedback resistor so that b¼ –106 V/A. In addition,the required
phase-sensitive detection is carried out using acomputer-based
lock-in amplifier40–42 consisting of an inex-pensive data
acquisition (DAQ) device and a LabVIEW soft-ware program. The
central features of this computer-basedlock-in are as follows:
First, triggered by the negative-goingtransition of a function
generator’s TTL (or square-wave)sync output, the DAQ device
acquires N (a power of two)samples of the I-to- V circuit’s voltage
output. The samplingrate is chosen to be fsampling ¼ Npoint � f ,
where Npoint is thenumber of samples to acquire during one
reference cycleand f is the reference frequency. A total of Ncycle
referencecycles are acquired so that the total number of
acquiredvoltage samples is N ¼ Npoint � Ncycle. Since the
digitizingprocess is triggered at the our defined zero-phase angle,
insoftware we create two copies of this acquired data wave-form and
multiply one copy by the “in-phase” reference2 sinð2pftÞ and the
other copy by the “quadrature” reference2 cosð2pftÞ. A fast Fourier
transform is then taken of eachof these arrays (hence the reason N
is chosen to be a powerof two) and the dc components of each picked
out, resultingin the in-phase and quadrature voltage amplitude at
frequencyf in the original digitized waveform. Thus, the
frequencybandwidth of our output signal is on the order of the
FFT’sfrequency resolution39 Df ¼ fsampling=N. We define the
timeconstant of our lock-in algorithm to be s � 1=Df , sos ¼
N=fsampling ¼ ðNpointNcycleÞ=ðNpointf Þ ¼ Ncycle=f . Figure
7illustrates how this lock-in algorithm is programmed
inLabVIEW.43
We will describe how to carry out the above-describedscheme with
two different commonly used low-cost DAQdevices: the USB-6009 and
the myDAQ.44 In each case, themanner in which the scheme is
implemented must be adaptedto the limitations of the DAQ
device.
1. Setup using the USB-6009
The USB-6009 device performs 14-bit analog-to-digitalconversions
of an incoming signal at rates up to 48,000Samples per second
(S/s). Each N-sample acquisition can be
Fig. 7. LabVIEW code to carry out two-phase lock-in amplifier
algorithm. A fast Fourier Transform performs the required low-pass
filtering to obtain the dc
value, which is the zero-index element in the array output by
the FFT.vi icon.
201 Am. J. Phys., Vol. 82, No. 3, March 2014 Reynolds, Panda,
and Essick 201
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-
hardware triggered by a TTL signal. The
device’sprogrammable-gain amplifier allows for eight possible
inputvoltage ranges over which to spread the 14-bit
resolution.Since our input signal is on the order of a few hundred
milli-volts, we choose the (device’s most sensitive) 61 V
range.Additionally, given the 48 kS/s maximum sampling rate andthat
fact that we need the number of samples per cycle(Npoint) to be a
power of two, we chose our reference fre-quency to be 1000 Hz.
Then, Npoint can be 32, the minimumvalue we feel necessary to
properly describe the 1000-Hzsignal. Finally, this DAQ device
possesses only modest digi-tal-to-analog conversion capabilities.
With a maximum ana-log output update rate of 150 Hz, the USB-6009
cannotproduce the reference signal we require for our
experiment.Hence, we retain the USB-interfaced Agilent 33210 A
func-tion generator for this setup, as shown in Fig. 8.
With f¼ 1000 Hz, we chose Npoint¼ 32 and Ncycle¼ 512.Then,
fsampling¼ 32,000 S/s, N¼ 16,384 (¼214), and s¼ 0.51 s.Using these
parameters, the room-temperature capacitance ver-sus reverse bias
data was acquired on a STPS20120D diode.The resulting 1/C2 vs VR
plot is shown in Fig. 9. Thestraight-line character of this plot
indicates that the diode’sdoping density is constant over the
spatial region profiled.Using Eqs. (6) and (7), the slope and
y-intercept of this plotdetermine that the doping density in this
region and the diode’sbuilt-in potential are 2.6� 1015 dopant/cm3
and 0.613 V,respectively. These results are in excellent agreement
with theresults obtained using research-grade instrumentation.
As with the research-grade setup, the uncertainties in ourq and
Vbi values were estimated by first comparing theresults of ten
identical runs of the experiment. We found thatsuch random errors
contributed uncertainties in m and b onthe order of only 0.5%. So,
again, the uncertainty in the con-tact area is the dominant
contribution to dq, yieldingdq¼ 0.1� 1015 dopant/cm3. With dm=m �
db=b � 0:005,we predict dVbi� 0.004 V.
Using these same CV data, Eqs. (16) and (17) yield the q(W)vs W
plot shown in Fig. 10. This plot indicates constant doping
density over the region profiled and again is in excellent
agree-ment with the research-grade results (Fig. 6). We note that,
asdescribed by Eq. (21), the doping density determinationbecomes
more noisy as the profile probes deeper into the sam-ple. This
noise is more noticeable in Fig. 10 than in Fig. 6because dC is
larger for the USB-6009 setup. From the data inFig. 10, we
determine that in the range W¼ 2.00–2.25 lm,dq¼ 1.6� 1014
dopant/cm3. Using this value, along with theother known quantities
in Eq. (21), we find that dC¼ 0.07 pF isresponsible for the
observed scatter in q.
2. Setup using the myDAQ
The myDAQ device performs 16-bit analog-to-digital con-versions
of an incoming signal at rates up to 200,000 S/s.The device’s
programmable-gain amplifier allows for twopossible input voltage
ranges. We choose the (device’s mostsensitive) 62 V range. However,
the device offers no hard-ware triggering capability for these
digitizing operations.Thus, the triggering for the voltage
acquisitions must bedone in software. Finally, this DAQ device can
also perform
Fig. 8. Low-cost implementation of capacitance profiling using
USB-60009
DAQ device. The op-amp (LF411) circuit serves as a current
preamplifier
and the lock-in algorithm is carried out using a
hardware-triggered DAQ de-
vice and LabVIEW software. Because the DAQ device has minimal
wave-
form generation capabilities, a stand-alone computer-interfaced
Agilent
33210 A function generator is used.
Fig. 9. Experimental data (1/C2 vs VR) obtained at room
temperature on aSTPS20120D Schottky diode using USB-6009 DAQ device
(time constant
of lock-in is s¼ 510 ms). This plot determines the diode’s
constant dopingdensity and built-in potential to be 2.6� 1015
dopant/cm3 and 0.613 V. Errorbar for each data point is smaller
than symbol used to represent point.
Fig. 10. Spatial profile of phosphorous doping obtained using a
USB-6009
DAQ device. A constant value for q(x) of about 2.6� 1015
dopant/cm3 isindicated over the spatial range W¼ 0.5–2.3 lm.
202 Am. J. Phys., Vol. 82, No. 3, March 2014 Reynolds, Panda,
and Essick 202
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-
digital-to-analog conversions at rates up to 200,000 S/s ontwo
analog output (AO) channels. We use these two AOchannels, each
operating at 200 kS/s, to produce our requiredmodulated bias
voltage as well as a digital reference signal,whose transitions are
in-phase with the bias voltage’s acmodulation. As shown in Fig. 11,
a 60.5 V square-wave ref-erence signal generated by one of the
analog output channelsis directly connected to, and read by, one of
the analog inputchannels. The moment at which a transition of this
squarewave occurs is determined by searching the acquired wave-form
in software, enabling lock-in detection. Also, the modu-lated bias
voltage produced by the other AO channel ispassed through an
(inverting) op-amp low-pass filter in order
to suppress digital quantization. Figure 12 illustrates
howsoftware triggering is carried out in LabVIEW.
With f¼ 1000 Hz, we chose Npoint¼Ncycle¼ 128. Then,fsampling¼
128,000 S/s, N¼ 16,384 (¼214), and s¼ 0.13 s.Using these
parameters, the room-temperature capacitanceversus reverse bias
data was acquired on a STPS20120Ddiode. The resulting 1/C2 vs VR
plot is shown in Fig. 13.Again, the straight-line character of this
plot indicates thatthe diode’s doping density is constant over the
spatial regionprofiled. Using Eqs. (6) and (7), the slope and
y-intercept ofthis plot determine that the doping density in this
region andthe diode’s built-in potential are 2.6� 1015 dopants/cm3
and0.623 V, respectively. These results are in excellent agree-ment
with those obtained using the other setups.
As before, the uncertainties in our q and Vbi values
wereestimated by first comparing the results of ten identical
runsof the experiment. We found that such random errors
con-tributed uncertainties in m and b on the order of 0.6%.
Thus,
Fig. 11. Low-cost implementation of capacitance profiling using
a myDAQ
DAQ device. The op-amp (LF411) circuit serves as a current
preamplifier
and the lock-in algorithm is carried out using a
software-triggered DAQ de-
vice and LabVIEW software. The waveform generation function of
the
myDAQ device is used to create a modulated reverse bias. The
op-amp low-
pass filter with f3dB� 5 kHz suppresses digitizing steps on
small-amplitudeac modulation.
Fig. 12. LabVIEW code to carry out software analog
triggering.
Fig. 13. Experimental data (1/C2 vs VR) obtained at room
temperature on aSTPS20120D Schottky diode using a myDAQ DAQ device
(time constant
of lock-in is s¼ 130 ms). This plot determines the diode’s
constant dopingdensity and built-in potential to be 2.6� 1015
dopant/cm3 and 0.623 V. Errorbar for each data point is smaller
than symbol used to represent point.
203 Am. J. Phys., Vol. 82, No. 3, March 2014 Reynolds, Panda,
and Essick 203
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-
dA is the dominant contribution to dq, yieldingdq¼ 0.1� 1015
dopant/cm3. With dm=m � db=b � 0:006,we predict dVbi� 0.005 V.
Using these same CV data, Eqs. (16) and (17) yield theq(W) vs W
plot shown in Fig. 14. Again, we see that, perEq. (21), the doping
density determination becomes morenoisy as the profile probes
deeper into the sample. This timethe noise is significantly larger
than in Figs. 6 and 10, indicat-ing that dC is larger for the myDAQ
setup in comparison tothe two other setups used. From the data in
Fig. 14, we deter-mine that in the range W¼ 2.00–2.25 lm, dq¼ 4.4�
1014dopant/cm3. Using this value, along with the other
knownquantities in Eq. (21), we find that dC¼ 0.2 pF is
responsiblefor the observed scatter in q. This larger uncertainty
in the ca-pacitance measurement (in comparison to the other two
set-ups) most likely derives from the lower-level of
precisioninherent in the software triggering used here.
IV. CONCLUSION
A summary of our results is given in Table I. Here, wesee
excellent agreement between the three setups, with pos-sibly slight
systemic shifts due to the different lock-indetection scheme,
cabling, and breadboard wiring used ineach case.
In conclusion, using research-grade instrumentation, wehave
demonstrated the capacitance profiling method and
verified its results are consistent with another
characteriza-tion technique (secondary ion mass spectroscopy).
Further,we have shown that, in spite of their limitations such as
re-stricted analog-to-digital conversion speed, inadequatewaveform
generation capabilities, and lack of hardware trig-gering,
inexpensive DAQ devices can be used to accuratelycarry out
capacitance profiling on semiconductor samples.These low-cost
solutions make the introduction of suchmeasurements an attractive
option for advanced laboratoryprojects.
ACKNOWLEDGMENTS
This work is dedicated to the memory of Professor J.David
Cohen.
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Table I. Summary of results on STPS20120D Schottky diode.
Setup q (�1015 dopants/cm3) Vbi (V) dC (pF)
Research-grade 2.7 6 0.1 0.622 6 0.001 0.02
USB-6009a 2.6 6 0.1 0.613 6 0.004 0.07
myDAQb 2.6 6 0.1 0.623 6 0.005 0.2
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