Modulation calorimetry in diamond anvil cells II: Joule-heating design and prototypes

8 MB·

by  Zachary M. Geballe

Modulation calorimetry in diamond anvil cells II: Joule-heating design and
prototypes
Zachary M. Geballe,1 Viktor V. Struzhkin,1 Andrew Townley,2 and Raymond Jeanloz3
1)Geophysical Laboratory, Carnegie Institution for Science, Washington, DC 20015
2)Department of Electrical Engineering and Computer Sciences, University of California, Berkeley,
CA 94720
3)Department of Earth and Planetary Science, University of California, Berkeley,
CA 94720
(Dated: 10 January 2017)
Part I shows that quantitative measurements of heat capacity are theoretically possible inside diamond anvil
cells via high-frequency Joule heating (100 kHz to 10 MHz), opening up the possibility of new methods
7
1 to detect and characterize transformations at high-pressure such as the glass transitions, melting, magnetic
0 orderings, or the onset of superconductivity. Here we test the possibility outlined in Part I, using prototypes
2 and detailed numerical models. First, a coupled electrical-thermal numerical model shows that specific heat
n of metals inside diamond cells can be measured directly using ∼ 1 MHz frequency, with < 10% accuracy.
a Second, we test physical models of high-pressure experiments, i.e. diamond-cell mock-ups. Metal foils of 2
J to 6 µm-thickness are clamped between glass insulation inside diamond anvil cells. Fitting data from 10 Hz

7 to ∼30 kHz, we infer the specific heat capacities of Fe, Pt and Ni with ±20 to 30% accuracy. The electrical
test equipment generates -80 dBc spurious harmonics which overwhelm the thermally-induced harmonics at

] higher frequencies, disallowing the high precision expected from numerical models. An alternative Joule-
i
c heating calorimetry experiment, on the other hand, does allow absolute measurements with <10% accuracy,
s √
despite the -80 dBc spurious harmonics: the measurement of thermal effusivity, ρck (ρ, c and k being
-
l density, specific heat and thermal conductivity), of the insulation surrounding a thin-film heater. Using a
r
t ∼50 nm-thick Pt heater surrounded by glass and 10 Hz to 300 kHz frequency, we measure thermal effusivity
m
with ±6% accuracy inside the sample chamber of a diamond anvil cell.
.
t
a
m
I. INTRODUCTION neering methods have been employed to study heat ca-
- pacity in-situ. The highest pressure experiments3–8 have
d
usedlaserheatingorresistiveheatingatfrequenciesupto
n High-frequency calorimetry of metal samples in dia-
hundredsofHzoreven10kHzinonecase,samplesrang-
o mond anvil cells has the potential to reveal Debye tem-
ing from nanoliters to microliters in volume, and maxi-
c peratures, deviations from Debye models, heat capacity
[ mum pressures and temperatures up to 13 GPa and 20
anomalies at magnetic, superconducting, or amorphiza-
K in one case4 and 0.3 GPa and 300 K in another.5
1 tiontransitions,andthelatentheatsofmeltingandother
v first-order transitions. Such measurements would com-
3 plement existing structure-sensitive high-pressure tech-
7 niques (e.g. x-ray diffraction) and enable comparison To make quantitative measurements at higher pres-
8
of diamond-cell data with shock-wave data that are in- sures and temperatures (10 GPa and 300 K to 100 GPa
1
trinsically adiabatic, but which operate under different and 3000 K), Part I of this two-part publication shows
0
. conditions (e.g. short time scales, high strain rates, irre- that even higher frequencies (kHz to MHz) are required,
1
versibility of pressure-temperature paths). and that Joule-heating enables absolute measurement of
0
Theprimarychallengeinsuchanexperimentistoheat specific heat. Here we extend Joule-heating modula-
7
1 a small sample in a nearly-adiabatic manner despite the tion calorimetry to higher frequencies and smaller sam-
: factthatitisborderedbyasolidorliquidofhighthermal ple sizes than previously achieved, using two methods
v
conductivity(k ∼1to30Wm−1K−1)andiswithin∼10 to study measurement accuracy: a detailed electrical-
i
X µm of diamond anvils (k ∼ 2000 Wm−1K−1), yielding a thermal model of the design introduced in Part I, and
r thermal diffusive timescale of ∼10 to 100 µs. laboratorymeasurementsofmetalheatersranginginvol-
a In fact, this challenge is also encountered in measure- ume from ∼ 4 to 40 picoliters that are pressed between
ments of materials grown on thermally conductive sub- glass insulation inside diamond anvil cells.
stratessuchasSiorAl O ,meaningtheresultsandanal-
2 3
ysispresentedheremayfacilitatemeasurementsinappli-
cations outside high-pressure research. In particular, if In both the numerical model and laboratory measure-
an as-grown material is > 1 µm thick, it is amenable ments, power is deposited via Joule heating of metal
to the same high-frequency calorimetric measurements foils and temperature oscillations are measured via the
studiedhere, withoutthe needforhigh-frequency modu- thirdharmonictechnique9 usingabridgecircuitadapted
lated lasers and photodiodes (as in Refs. 1 and 2). from one used at ambient pressure for specific heat
Within high-pressure experimental science, a few pio- spectroscopy.10
2
(a) 100 Ω 1 kΩ
Z=50 Ω
V
compensate

l
ca Ω Vout
i k In-amp
tr 1 ref
c
e
l
E
Z=50 Ω
47 Ω Sample
V
drive
R = R (1+αΔT) for ΔT= ¼ (V(A)+V(B)+V(C)+V(D))
sam sam,0

Sample
l Insulator
a A B C D
m
r
e
h
T I = I 2 R /4
source electrical sam
FIG. 1. Schematic of the coupled electrical-thermal model. (Top) Electrical designof bridge circuit, with red box highlighting
sample resistor. (Bottom) One-dimensional thermal model of half the sample thickness and one side of insulation, where
electrical symbols are used to model heat flow (as electrical current) and temperature (as voltage).
II. NUMERICAL METHOD resistor equals the main component of voltage across the
sample. We implement this model in LTSpice.12
First, we test whether a coupled electrical-thermal At ideal tuning, the bridge is “balanced” and most of
model predicts the same results as the thermal model thevoltageattheinvertinginputofthein-ampis“nulled
of Part I, in which temperature was simulated and used out” by the compensating voltage. An example illus-
to estimate electrical resistance, but with no feedback tratestuningofthebridge; Fig. 2showsthatat100kHz
from resistance to heating power.11 These coupled mod- frequency and ±4.8 V driving voltage, the voltage dif-
elstestafewkeyassumptionsimplicitinPartI:(1)that ference between midpoints of the two arms of the bridge
theamplitudesofcurrentsandvoltagesneededtoinduce can be minimized by balancing the bridge, resulting in a
measurablethirdharmonicvoltageoscillationsareinthe mustache-shaped waveform at the output of the in-amp
typical range available from commercial test equipment, (final frame of Fig. 2). Appendix A describes the cal-
(2)thattheresistanceoscillationsinourdesignaresmall culation of compensating voltage needed to balance the
enough to use the approximation “ 1 ≈1−x” to infer bridge. Alternatively, balance can be achieved by trial
1+x
heatcapacityinAppendixC,and(3)thatharmonicdis- and error.
tortions from the instrumentation amplifier do not bias
To model the temperature oscillation, the LTSpice
the third-harmonic temperature measurement.
electrical software is used once again. This time, elec-
Aschematicofourcoupledelectrical-thermalmodelis
trical components are used to make the elements of a
shown in Fig. 1. Two outputs of a waveform generator
finite element model that matches the one used in Part
drivevoltageoscillationthroughthetwoarmsofabridge
I of this two-part publication, with one exception: LT-
circuit, one of which contains the metal sample. The
Spice’s native time-stepping routine is used instead of
driving voltage in the sample arm is,
the Crank-Nicholson scheme used previously. Material
V sin(ωt) properties and dimensions of the sample assumed here
d
are identical to those modeled in Part I, matching the
Thegenerator’sotheroutputsendsacompensatingvolt- properties of iron at ambient conditions. The insulation
age, V sin(ωt), through two resistors, with an amplitude material is assumed to have the properties of silica glass
c
thatistunedsothatthevoltageacrossthe1kΩreference at ambient conditions, which is slightly less thermally
3
Sample (Fe) Insulator (silica glass)III. LABORATORY METHOD
d: Layer thickness (µm) 5 12.5
w: Width (µm) 20 20
To test whether our electrical and thermal design can
l: Length (µm) 100 100
be implemented in practice, we first study iron, plat-
ρ: Density (g cm−3) 7.9 2.2
c: Specificheat(Jg−1 K−1) 0.45 0.83 inumandnickelfoilspressedinsidediamond-cellsatnear-
k: Thermalconductivity(W 80 1.2 ambientpressure.14 Wemakediamond-glass-metal-glass-
m−1 K−1) diamondsandwichesusingtwopiecesofmicroscopeslide
r: Resistivity (Ω m) 9.7×10−8 0 coverslips (120 µm-thick silicate glass) and 2.4, 5.7 and
α: Temperature coefficient 0.0064 0 6µm-thickfoilofplatinum, ironandnickel, respectively.
of resistance (K−1) Theironandplatinumsamplesarecutwithrazorblades
while the nickel sample is laser-cut. By testing these
TABLEI.Propertiesofthesampleandinsulatorusedinour
circuits with electronics that mimic the schematic used
numerical models.
in our numerical model (Fig. 1), we determine whether
ourdiamond-cellcalorimetrydesignisfeasible. Potential
pitfallsincludeelectricalnoise,spuriousharmonicdistor-
Thermal Electrical tion,contactresistance,orelectromigrationofheaterma-
power, p(W) current, I(A) terial. We will show that spurious harmonic distortions
heat, Q(J) charge, Q(C)
limit our accuracy, but note that contact resistance did
temperature, T(K) voltage, V(V)
overwhelm the sought-after third harmonic during test-
heat capacity, c ρ A∆z capacitance, C(C2/J)
sam sam ingnotpresentedhereinthecaseofsilverepoxycontacts
(J/K)
cured at room temperature.
thermal resistance, ∆z/kA resistance, R(Ω)
(K/W) Second, we test a thin-film of platinum (50 nm thick)
sputtered onto the central 16±1 µm-wide region of 10
TABLEII.Thermalparametersandtheelectricalparameters µm-thickglassdisc,usingphotolithography. Itispressed
used to model them, including variable and units. Note that against a second disc of glass (20 µm-thick) inside the
A is the sample surface area, and ∆z is the thickness of each sample chamber of a diamond-cell at near-ambient pres-
element in our finite element model. sure. Heater length and width are measured with an
optical microscope, while thickness is measured with a
Zygo optical surface profiler.
Theelectricaltestequipmentisthesameforboththin-
conductive than the KBr insulation modeled in Part I. film and foils, and follows the same design used in our
We choose to model a different insulating material here numerical models (Fig. 1). A 14-bit, 500 megasamples
in order to decrease the addenda contribution to total per second waveform generator (BK Precision 4065) de-
heat capacity and to enable comparison with laboratory liversbothdriveandcompensationsinewavesviaitstwo
tests using a metal film deposited on glass (see below). outputs. Resistorsandtheamplifierforthebridgecircuit
Fig. 1 outlines the thermal model used in LTSpice. are soldered onto the homemade circuit board shown in
The sample is divided into four elements, each of which Fig. 3b. The amplifier is powered by ±10 V DC power,
is heated by a current source equal to one-forth the elec- with 1 µF bypass capacitors to ground that filter out
trical power deposited in the sample. This flow of heat high-frequency noise. The output from the amplifier is
increases the temperature of the sample elements, which readbyan8-bit500megasamplepersecondoscilloscope
arelinkedviathermalresistances,andconnectedtoase- with minimum sensitivity of 2 mV per division (Lecroy
quence of twenty insulator elements that are also linked LT342), or by a 10 MHz lock-in amplifier (Zurich In-
viathermalresistances.13Correspondencesbetweenther- struments HF2LI). A third alternative is to avoid the
mal parameters and electrical parameters used in the in-amp and to compensate the sample’s first harmonic
computer model are listed in Table II, and details are voltage through the differential input of the lock-in am-
given in Appendix B. plifier instead. All three voltage-measuring schemes are
tested, and the differences are seen to be negligible (sec-
The thermal and electrical models of the sample are
tion VI).15
coupled in the following way. Electrical power across
the electrical model of the sample causes heat flow in
the thermal model of the sample: dQ/dt = I V /4,
sam sam
whereI andV arevaluesofcurrentacrossthesam- IV. NUMERICAL RESULTS
sam sam
ple, Q is the heat added to each sample element, and
the 4 accounts for the four sample elements. The aver- Fig. 2 shows several consequences of a 100 kHz driv-
agetemperatureofthefourthermalsampleelements, T, ingvoltagefortheironheatermodelednumerically. The
modulates the resistance of the electrical model of the quantity we measure in laboratory experiments is shown
sample: R =R (1+αT) where R =r·l/w·d is the in the final figure: the amplitude of third harmonic volt-
sam 0 0
room temperature sample resistance and α = dlogr/dT age oscillation, V .
3ω
is the temperature coefficient of resistance. Appendix C shows that by measuring V , along with
3ω
4
100
302.88
A) K)
(m 0 T ( 302.86
I
−100 302.84
10
98.79
)
Ω
V) m
m 0 ( 98.78
( m
V sa 98.77
R
−10
1 1.02
W) 0.8 mV) 1.01
0.6
m (
( 0.4 ut 1
p Vo
0.2
0.99
0 5 10 0 5 10
time (µs) time (µs)
FIG. 2. Result of numerical model at 100 kHz frequency. From top-left, a 100 kHz, 100 mA alternating current drives 10
mV voltage oscillations in the sample, creating a 200 kHz power oscillation from 0 to 1 mW, which causes the average sample
temperature to oscillate from 0.69 to 0.73 K above room temperature, causing resistance to oscillate from 97.76 to 97.79 mΩ.
Thisdynamicresistancefeedsbackintovoltageoscillationsacrossthesample,whichcanbemeasuredwiththehelpofabridge
circuit (Fig. 1) that nulls out the dominant signal due to the average value of resistance. The residual voltage is shown in the
bottom-right panel. It is the output of an in-amp, which amplifies the difference in voltage across the bridge, with a gain of
10.9. It also distorts the signal and adds a DC offset, but the in-amp does not generate significant spurious voltages at the
third harmonic frequency. Dashed curves show what the temperature, resistance and residual voltages would be in the ideal
case of no addenda contribution and ideal electronics.
thevaluesoftime-averagedresistance,R ,andin-amp We now apply these formulas to the numerically mod-
sam
gain, G, we can experimentally determine the amplitude eled third harmonic amplitudes. The temperature oscil-
of temperature oscillations in the heater (also see Ref. lation inferred from the measurement of third harmonic
10): amplitude, V =7.8 µV, would be T =22 mK, which
3ω 2ω
is 11% smaller than what the temperature oscillation
would be in a truly adiabatic experiment. The inferred
2(R +R )2V
T = b sam 3ω (1) heat capacity would be Ctotal = 39 nJ/K, which is 11%
2ω αVdRsamRbG larger than the 36 nJ/K heat capacity of the 79 ng mass
(10picolitervolume)ironsamplemodeledhere. Thedis-
where R = 97 Ω is the buffer resistance (i.e. total
b crepancy arises from heat loss to the addenda (i.e. ther-
resistance between voltage-generation and sample), α is
malconductionintotheinsulationthatincreasesthespa-
theassumedormeasuredtemperaturecoefficientofresis-
tial extend, hence total heat capacity of the dynamically
tance, and V is the driving voltage inside the waveform
d heated region).
generator(i.e. thenominalvoltageinthe“highZ”mode
of the BK 4065 waveform generator).
The frequency dependence of this inferred heat capac-
Since we also know the power deposited via Ohm’s
ity is shown in Fig. 4. At frequencies ≤ 1 MHz, the re-
law, we can determine the heat capacity of the heater
sultsofthiscoupledelectrical-thermalmodelmatchthose
plus addenda (i.e. whatever nearby insulating material
ofthethermalmodelpresentedinPartI,confirmingthat
is dynamically heated):
the voltage oscillations resulting from calculated ther-
mal oscillations approximate those in the more-realistic
case of coupled voltage and thermal oscillations. The
C =p /2ωT
total 2ω 2ω match also shows that at low-enough frequencies, a real-
1 V 1
= ( d )2R istic in-amp provides a high-fidelity voltage output that
2 Rsam+Rb sam2ωT2ω can be used to infer heat capacity of a metal sample.
αV3R2 R G At frequencies beyond 1 MHz, however, the in-amp dis-
= d sam b (2)
8ω(R +R )4V tortsthethirdharmonicmeasurement,suggestingthatat
sam b 3ω
5
FIG. 4. Total heat capacity that would be measured in nu-
merical models, normalized by heat capacity of the sample
alone, as a function of frequency. Colors indicate the one-
dimensional finite element model that was used: thermal
model described in Part I (blue), or the coupled electrical-
thermal model described here with ideal models (pink) or
FIG.3. (a)Astripofironrestingona120µm-thickpieceofa
manufacturer supplied models (green) of the in-amp. Yellow
glassontopofadiamondanvil,withcopperleadsrestingon
highlights the region of <10% error. Note that frenquencies
top of the iron. (b) The same materials as in (a), but after a
plottedinthispaperrefertothefundamentalmodeofcurrent
second glass-covered diamond anvil has been pressed against
and voltage oscillation, and are therefore two-times smaller
the iron and copper. (c) The circuit board used to connect
thantheheatingfrequenciesreferredtoas“frequency”inPart
electrical test equipment (wavefunction generator, DC power
I.
supplyandoscilloscope)totheironsample. (d)Diamondcell
connected to electrical board.
) −4
V
least with this differencing amplifier, high-fidelity third m
−6
harmonicvoltagescannotbeextractedatthehighestfre- (

e
quencies studied in Part I of this two-part publication. g −8
a
t
ol −10
V
−12
V. LABORATORY RESULTS
0 0.5 1
Time (ms)
An example of laboratory data is shown in Fig. 5. As
expected,thevoltagemeasuredacrossthebridgeincludes
FIG. 5. Measured voltage timeseries (dots) and curve fitted
two components of roughly equal magnitude: a first har-
usingfirstandthirdFouriercomponents. Fivehundred1ms
monic and a third harmonic. The third harmonic ampli-
cycles were averaged to produce the timeseries.
tude, 4.1 mV, is used to infer a 1.1 K temperature oscil-
lation following Eq. 1, with all values of R equal to
sam
the resistance of the sample section of the bridge circuit,
R2pt, except for the denominator, where R = R4pt, the surroundings, and is defined by:
sam sam sam
the resistance of the sample alone, not including contact
p
and lead resistance.16 K = 2ω (3)
th T
2ω
We also vary frequency and driving voltage and plot
the resulting third harmonic amplitudes and tempera- where p = 1I2R4pt and I = V /(R +R2pt ). In
2ω 2 ω sam ω d b sam
ture oscillations in the first two panels of Fig. 6. We other words, it is the total heat capacity, C , times
total
then calculate the resistance of the system to changes in the heating frequency, 2ω.
temperature, hereafter referred to as “effective thermal The top three panels of Fig. 6 show how measured
conductance” and denoted K . It accounts for both the third harmonics of voltage, inferred temperature oscilla-
th
heat capacity of the metal sample and conductance to tions, and effective thermal conductances vary with fre-
6
FIG. 7. Same as Fig. 6, but with the platinum foil replacing
FIG.6. Laboratorydataandfittedheatcapacitiesofaniron
the iron sample, slightly different driving voltages (±10, 8.2,
sample. Thetopthreepanelsshowmeasuredthirdharmonic
6.8 and 5.6 V for blue, green, red and cyan circles) and 100
amplitudes, inferred temperature oscillations, and effective
kHz bandwidth used for the fit shown in the third panel.
thermalconductanceofthesampleplussurrounding,plotted
versus the frequency of driving voltage, ω/2π. Blue, green,
red and cyan indicate ±9.6, 7.8, 6.4, and 5.4 V driving volt-
ages, respectively. Shaded areas reflect uncertainties. The quency or decreasing driving voltage, temperature oscil-
solidblackcurveinthethirdpanelshowsthethree-parameter
lations decrease and so do third harmonic voltages. The
fitusingdatafrom10Hzto20kHz,whiledashedblacklines
fact that the effective thermal conductance, K , is in-
show the contributions to K from each of the three terms th
th dependent of driving voltage, means that a true third-
in the model. Black circles in the bottom two panels show
harmonic is being measured (i.e. V ∝V3).
total sample heat capacity and specific heat of iron inferred 3ω d
from model fits that use data from 10 Hz to maximum fre- The shape of the K vs. frequency curve can be un-
th
quenciesof2kHzto20kHz,withuncertaintiesshowningrey. derstood using a one-dimensional model of heat flow.17
The dashed black line in the bottom panel shows the litera- We model K , the heating rate required to raise the
th
turevalueofspecificheatandyellowshadesthevalueswithin
temperature of the metal by one degree, as the sum
±10% of the literature value.
of three rates: (1) the rate required to raise the tem-
perature of the metal alone, c ρ Vol · 2ω, (2) the
sam sam
rate required to heat the insulation via thermal diffu-
√ √
sion, k ρ c ·Area· 2ω, and (3) the rate required
ins ins ins
quency and driving voltage. Since we show in section tomaintainalineartemperaturegradienttothediamond
VI that uncertainties in V3ω are approximately -80 dBc heatsink,kinsArea/dins. Thesethreecontributionstothe
(i.e. 0.01% of the voltage across the sample, multiplied effectivethermalconductanceareshownasblackdashed
by gain), we assume this value here, and propagate it, lines in Fig. 6, with slopes of 1, 1 and 0 in log-log space,
along with 15% uncertainty in Rs4apmt, to uncertainties in and with their sum (i.e. the fit t2o data) shown as a solid
T2ω and Kth. black curve. Fitted parameters in this one-dimensional
Asheatinputpercycledecreasesduetoincreasingfre- model provide qualitative measures of insulation thick-
7
Theresultsofthesameanalysisonplatinumandnickel
foilsofsimilardimensionsshowsimilarresults. A350ng-
mass strip of platinum (460×15±1.5×2.4±0.2 µm) is
foundtohaveaspecificheatof29±5J/molK(compared
to the literature value of 26 J/mol K) by fitting data
from 10 Hz to 100 kHz. Here, the higher bandwidth is
enabledbytheslightlysmallercross-sectionalareaofthis
foil compared to the iron foil. Measurements of a lower
resistance strip of nickel almost matches the literature
value within the uncertainty: by fitting data from 10 Hz
to10kHz, a350ng-masspieceofnickel(420×16±1×6
µm) is found to have a specific heat of 20±5, whereas
the literature value is 26 J/mol K.18
We also test an alternative Joule-heating calorimetry
experiment: measurementofthermaleffusivityofthein-
sulation surrounding a thin-film heater. The results are
shown in Fig. 9, with the same data processing for T
2ω
and K as in the case of metal foils. The model used
th
to fit K , however, is slightly different. Rather than
th
using a three parameter fit and interpreting the heat ca-
pacity term quantitatively, we assume the heat capacity
term is negligible and that the thermal effusivity term is
quantitatively meaningful. Despite not modeling the ex-
perimenthere,weexpectthisproceduretogivequantita-
tiveestimatesofthermaleffusivitysinceasfrequencybe-
comes large, the one-dimensional model mimics the real-
ityofheatdiffusingfromathin-filmheaterintotheglass
insulation. More precisely, our one-dimensional model is
realistic when the heating timescale is short compared
to timescale of thermal diffusion across the width of the
thin-film heater, width2c ρ /k , but long compared
ins ins ins
FIG. 8. Same as Figs. 6-7, but with a nickel foil instead tothetimescaleofconductionoutofthemetalthin-film,
ofironorplatinumsample, slightlydifferentdrivingvoltages (d c ρ )2/(c ρ k ). For the experiment
heater heater heater ins ins ins
(±10, 9, 8 and 7 V for blue, green, red and cyan circles) and tested here, this requires the heating frequency, 2f, to
7 kHz bandwidth used for fit shown in the third panel. fall within the range 2.7 kHz<2f <110 MHz.
Indeed, the final panel of Fig. 9 shows that by using
data from 10 Hz to any maximum frequency between 10
kHz and 300 kHz, the fitted value of thermal effusivity
nessandeffusivity,butaquantitativelymeaningfulvalue
matchestheliteraturevaluewithinthe±6%uncertainty.
of sample heat capacity since the heat-capacity term is
Theprecisionofthefittotheproductofthermaleffusiv-
most sensitive to data at high-frequency, where the one-
ity and surface area is better, reaching 2.5% when data
dimensional model mimics reality. The fit to K data
th up to 300 kHz is used.
from 10 Hz to 24 kHz at all driving voltages yields a to-
tal heat capacity of 74±19 nJ/K, shown as a black dot
withgreyerrorenvelopeat24kHzinthefourthpanelof
Fig 6. Fits are weighted using measurement uncertainty, VI. ERROR ANALYSIS
and determined (along with covariance) via the “curve
fit”functionwithintheSciPylibraryinpython. Wealso To estimate the uncertainty in measured third har-
test how the maximum frequency of fitted data controls monic amplitude, we measure background spurious har-
heatcapacityestimates(fourth panelofFig. 6)andspe- monicsintwoways: thirdharmonicsofadummysample
cific heat estimates (fifth panel). To estimate specific and second harmonics of the real sample. The dummy
heat and its uncertainty, we divide fitted heat capacity sample is a 1.5 Ω off-the shelf resistor (< 500 ppm/◦C
by the number of moles in the 530×8.5±1×5.7±0.5 temperature coefficient, 1/4 W power rating).
µm piece of Fe, and add the 20% uncertainty in sample Figs. 10-12 show that both estimates of spurious har-
volume to the total uncertainty (assuming uncertainties monics imply a spurious free dynamic range of 80±10
addinquadrature). Usingdatafrom10Hzto20kHz,for dBc. The plotted data is normalized by dividing by gain
example, weestimatespecificheattobe20±9J/molK, timesthevoltagemeasuredacrossthesample,V . Third
ω
which is consistent with the literature value of 25 J/mol harmonics measured during real experiments (same as
K. shown in Figs. 6-9) are also plotted for reference. They
8
show that the signal starts ∼ 30 dB above the spurious tion (hence limited bandwidth) in our experiments may
signalatlowfrequency, andapproachesthespurioussig- be the waveform generator, the voltage measuring de-
nal around 20 kHz for metal foils, whereas it approaches vices, or both. In fact, the harmonic distortion expected
the spurious signal around 600 kHz in the case of the in either differencing amplifier is in the range measured
thin-film. in our experiment: total harmonic distortion generated
The improved bandwidth for thin-films can be under- in the HF2LI under one-sided drive is, according to the
stood by the increase in thermally-induced third har- manufacturer,approximately-70dB,whilethethirdhar-
monic voltage for large values of V , the voltage os- monic distortion from the manufacturer of AD8429 (a
ω
cillation across the heater. We expect the thermally- newer version of AD8421) ranges from -90 dB to -60 dB,
inducedthirdharmonictobeproportionaltoV3 whereas dependingonfrequencyandgain.19Still,weexpectlower
ω
the spurious third harmonic are likely proportional to harmonicsunderthesymmetricdriveusedherethanthe
V in our setup. V , in turn, is limited to the value one-sided drive used to test the amplifiers, suggesting
ω ω
10V·R /(97Ω+R )≤150mVforthelow-impedance that the -80 dBc spurious signal may come from an al-
sam sam
metal foils (0.3 to 1.4 Ω), but it is much larger for the ternative source.
high-impedance thin-film (26 Ω). In practice, we limit The other likely source of spurious harmonics is the
the driving voltage to ±6.3 V (compared to the ±10 V waveform generator. The manufacturer reports < −54
possiblefromourwaveformgenerator),butthisstillgen- dB harmonic distortion, but this is reported as an up-
erates a 10-times larger voltage across the heater, com- per limit and it is measured from one channel, so we
pared to the metal foils. expect smaller distortions when nearly-balanced wave-
Usingtheironfoil,weperformthetestsofbackground forms from the two outputs are differenced, as in our
spurious harmonics for three different voltage-measuring setup. To discriminate harmonic distortions internal to
schemesmentionedabove: instrumentationamplifierfol- the voltage generating unit from those internal to the
lowedbyoscilloscope,instrumentationamplifierfollowed voltage measuring unit, lower-distortion test equipment
bylock-inamplifier,andthelock-inamplifieralone(using would be needed. In fact, analog audio analyzers may
its differential inputs). In all cases, we measure similar providelowerharmonicdistortionthatwouldenablesuch
harmonics, showing that a lock-in amplifier does not im- tests. For example, the Keysight U8903A is reported to
prove the ratio of signal to spurious signal, as expected havetotalharmonicdistortion≤−101dBat20Hzto20
forharmonicdistortionsgeneratedinsidepreamplifiersor kHz20. Use of such a device in a real experiment could
waveform generators. also improve measurement accuracy.
But harmonic distortion is not the only source of un-
certainty in our measurement. Uncertainty in sample
VII. DISCUSSION resistance is a key error source in the measurements of
metal foils because we only measured the two-point re-
Results of both the numerical and physical model sistance of the sample plus leads and contacts, leaving
of high-pressure experiments show a trade-off between uncertainty in the theoretically calculated sample resis-
bandwidth and measurement accuracy. In all cases, the tance. In fact, this ∼ ±10% uncertainty propagates to
trade-offiscausedbyspuriousharmonicsthatoverwhelm ±20% uncertainty in T2ω and Kth. In the case of a thin-
the third-harmonics induced from thermal oscillations in film heater with low contact resistance, the two-point
the sample. The coupled electrical-thermal numerical resistance measurement, Rs2apmt, is assumed to be an ac-
modelshowsthatthebandwidthlimitofourexperiment curate measure of the true sample resistance, Rs4apmt.
is at most 3 MHz, at which point the instrumentation Finally, the uncertainty estimate for specific heat or
amplifier starts to generate overwhelming spurious har- thermaleffusivityincludesasignificantcontributionfrom
monics. uncertaintyinheatersize. Hereweuselasercutting,cut-
The physical model of diamond-cell experiments re- ting by hand with a razor blade, and photolithography
vealsasignificantlymorelimitedbandwidthinthecaseof of a poorly adhered platinum film. These techniques re-
low-impedance heaters. Third harmonic measurements sult in somewhat rough edges, resulting in uncertainties
areoneorderofmagnitudeabovebackground(suggesting of ±1 µm or ∼ 10% in width, which propagates to 10%
10%electricalerror)at1to10kHzfor∼5×13×600µm in specific heat or effusivity.
metalstrips. Thelargerimpedanceoftheplatinumthin- In principle, the uncertainties in heater resistance and
filmallowsforsignificantlymoreaccurateelectricalmea- dimensions can be greatly reduced by use of advanced
surements at high frequency; the third harmonic mea- fabrication techniques such as focused-ion-beam milling,
surementisoneorderofmagnitudeabovebackgroundat mechanical cutting with a micromanipulator, or pho-
300 kHz. tolithography. Altogether, the improvements outlined
Nonetheless, by fitting data on metal foils from 10 Hz here could enable measurements with 1% to 10% accu-
to ∼ 10 kHz to a three parameter model, we find heat racy, as modeled numerically.
capacities that agree with the literature value to within But using the current setup with the current error
the ±20 to 30% uncertainty of the fit. sources,quantitativehigh-pressuremeasurementsofheat
The dominant source or sources of harmonic distor- capacity or thermal effusivity are already possible if the
9
appropriate heaters, samples and pressure devices are VIII. CONCLUSIONS
employed. The design must simply allow for the uncer-
tainties measured here: -80 dBc harmonic purity at ±10
Physical models of high-pressure Joule-heating
V driving voltage, and ±1 µm sample dimensions.
calorimetry experiments show that heat capacity can
be measured directly with 20 to 30% accuracy, while
In a diamond-cell capable of reaching 30 GPa (using
thermal effusivity can be measured with 6% accuracy.
anvilswith∼500µm-diameterculets),thiswouldenable
Harmonic distortions due to electrical test equipment
heatcapacitymeasurementsofmetalswithslightlylower
cause these uncertainties to be larger than those
accuracy than those presented here, because the sample
estimated from numerical models. Nonetheless, the
would have to be slightly shorter than the ones studied
current experimental setup may enable a wide variety of
here. Heatcapacitymeasurementsofsemimetalsorsemi-
experiments in high-pressure science.
conductors, on the other hand, would reach significantly
higher accuracy if they were shaped in a way that their
impedances were 10 to 100 Ω.
Appendix A
Themeasurementsofthermaleffusivitypresentedhere
are from samples loaded into the 300 µm-diameter sam-
We derive an approximate analytic solution to the
ple chamber of a diamond cell capable of reaching ∼ 20
steady state heat equation in order to calculate the re-
GPa. Hence,itispossiblethatdiamond-cellexperiments
sistance needed to balance the electrical bridge in our
to >10 GPa could reach the ±6% accuracy documented
coupled electrical-thermal numerical model.
here for thermal effusivity. One significant challenge in
The voltage source, V sin(ωt), drives current through
d
performing such high-pressure measurements is to avoid
a series of resistors with total initial resistance R , in-
ti
breaking the thin-film heater upon application of pres-
cluding the sample, which heats up. The sample’s tem-
sure.
perature increases to a higher steady state value, T ,
DC
and oscillates, T (t), causing an increased total resis-
Inalargervolumepresurecell,theuncertaintyinheat AC
tance, R +R α(T +T ), where R is the ambient
capacity could be significantly reduced, provided the ∼ ti 0 DC AC 0
temperature sample resistance and α=dlogR/dT is the
5×13µm-crosssectionalareacouldbemaintainedwhile
sample’s temperature coefficient of resistance. A ther-
increasing the length of metal sample. Specifically, we
mal link to a constant-temperature reservoir (e.g. the
expectuncertaintyinV tobeinverselyproportionalto
3ω
diamonds) with thermal conductance K (units: W/K)
length if all other parameters are fixed, in this case of a th
cools the sample. For simplicity in this steady state cal-
low-impedance heater. Alternatively, a thin-film heater
culation,weassumetheinsulation’sheatcapacityiszero,
(or other heater with ∼ 10 to 100 Ω impedance) would
meaning the change in temperature of the sample is due
allow quantitative measures of effusivity, as in the case
to two terms only: Joule heating, I2R, and heat con-
of diamond-cell experiments.
ducted away, K T. The current, I, is a ratio of driving
th
Analogous measurements of as-grown materials on voltage to time-dependent total resistance:
high thermal conductivity substrates should follow the
same guidelines: using the electrical test equipment em- I = Vdsin(ωt)
ployed here, ∼ 1 Ω Joule-heaters will allow measure- R +α(T +T )R
ti DC AC 0
ments up to ∼ 10 kHz, while 10 to 100 Ω heaters will
allow measurements up to 300 kHz frequency. Possible resulting in the following heat equation:
applications include direct measurement of heat capac-
ity of ∼ 5 × 13 × 600 µm-thick semiconductors grown
on sapphire substrates (e.g. GaN21), or as in the case C dTAC =I2R−K T
of the high-aspect ratio metal sample proposed above, sam dt th
a metal sample of similar cross-section and ten-times V2sin2(ωt)R (1+α(T +T ))
greater length (e.g. MgB222). = d (R +α(0T +TDC)R )2AC
ti DC AC 0
Finally, we note that the accuracy needs of measure- −Kth(TDC +TAC) (A1)
ments depend greatly upon the scientific or engineering
question to be addressed. For example, the 10% accu- .
racy threshold assumed here is not likely to be useful Tosolveforthesteadystatetemperature,weeliminate
for detailed thermodynamic analysis, but is sufficient for thesinusoidalterms,assumetemperatureoscillationsare
detection of a wide variety of second-order phase tran- small (TAC (cid:28) 1), and use the identity sin2(ωt) = 1(1−
TDC 2
sitions. Moreover, progress in sample preparation (to cos(2ωt)) to arrive at the expression:
reduceuncertaintiesinvolume)andelectricaltestequip-
ment (to increase bandwidth and/or decrease harmonic
distortion) may lead (cid:28) 10% accuracy in future heat ca- 0= 21Vd2R0(1+αTDC) −K T
pacity measurements. (R +αT R )2 th DC
ti DC 0
10
Dividing by −K and rearranging, rectsolution. Weconfirmthecorrectchoiceofcuibcroot
th
by running the numerical simulation itself.
V2R (1+αT )
0=T − d 0 DC
DC 2K (R +αT R )2
th ti DC 0
Appendix B
Multiplying by (R + αT R )2, expanding and
ti DC 0
groupingintopowersofT resultsinthefollowingcubic Here we derive the capacitances, resistances, and cur-
DC
equation: rent sources needed to implement our one-dimensional
heat flow model in LTSpice. Part I of this two-part pub-
lication explains the reduction of the heat equation to
(cid:18) V2αR (cid:19) one dimension in our planar model:
0=(αR )2T3 +2αR R T2 + R2 − d 0 T
0 DC 0 ti DC ti 2Kth DC ∂T 1 (cid:18) ∂2T (cid:19)
V2R = k +p
− d 0 ∂t ρc ∂z2
2K
th
where k, c, and ρ are material properties listed in Table
To find the steady state temperature rise, we use the 6.1, z is the axial direction in a diamond cell, and p is
cubic formula, and assume the maximal root is the cor- power density.
Discretizing, rearranging, and multiplying by A where A is the surface area of the metal sample, we find,
A
Tn+1−Tn 1 (cid:18)k A(T −T )−k A(T −T ) (cid:19)
i i = i+0.5 i+1 i i−0.5 i i−1 +p A∆z (B1)
∆t ρcA∆z ∆z i
wherethesubscriptmarksthepositioninspace(withfractionsindicatingthevalueofthelinkbetweentwoelements),
and the superscript marks the position in time.
In the LTSpice implementation of the electrical schematic shown in Fig. 1, the rate of change of voltage across
capacitor i at time n is current-in minus current-out, and current is given by the negative gradient of voltage divided
by resistance:
Vn+1−Vn I −I +Isource
i i = i−0.5 i+0.5 i
∆t C
i
(cid:18) (cid:19)
1 V −V V −V
= i−1 i − i+1 i−1 +Isource (B2)
C R R i
i i−0.5 i+0.5
where the fractional subscript i+0.5 indicates the resistor that separates capacitors i and i+1.
Comparison of Eqs. (B1) and (B2) shows that the Appendix C
analogies listed in Table II are valid:
V ←→T
i i Here we calculate the amplitude of temperature os-
Ci ←→ρcA∆z cillations, T2ω, and the heat capacity, Ctotal, to be in-
∆z ferred from a measurement of third harmonic voltage,
R ←→
i kiA V3ω. First we calculate the more intuitive relationship,
Isource ←→p A∆z V3ω as a function of T2ω, and then invert for the de-
i i
sired formula. Let R be the time-averaged resistance
sam
To improve accuracy, we average models using two al- of sample and let R be the other resistors in the sam-
b
ternative values of resistance at the interface between ple arm of the electrical bridge. Note that the sample’s
insulation and sample: R = 2∆z and R = resistance should be measured using a DC or first har-
inter kins+ksam inter
R . Thedifferencebetweentemperatureoscillationsin- monic measurement. In principle, α should also be mea-
ins
ferred from the two models is 20% over the range plot- sured by varying temperature slightly during a DC or
ted in Fig. 4, but the average value is within 2% of the other low frequency resistance measurement, though in
numerically-accuratethermalmodelofPartIofthistwo- practice we assume literature values for the foils of iron,
part publication. platinum and nickel.
The temperature oscillation causes a resistance oscillation in the sample:
R (t)≈R (1+αT sin(2ωt+φ))
sam sam 2ω