Thermopile Infrared Detector

1. Theory

• Definition: A thermopile is a thermal detector made of many thermocouples connected in series: the hot junctions lie on a thin absorbing membrane that is heated by the infrared radiation, the cold junctions lie on the substrate, and the Seebeck effect gives a voltage of $N{\alpha }_{12}\Delta T$.
• No bias: The thermopile produces a voltage by itself and is read without a bias current, so that its noise in the sources is the Johnson noise of its resistance; this is a statement of the sources’ noise model, not a measurement of the absence of other noise.
• Materials: CMOS-compatible thermopiles use n-type and p-type polycrystalline silicon, with aluminium as the contact; the measured Seebeck coefficients are of the order of -110 to -124 μV/K (n-type) and 106 to 137 μV/K (p-type), and the legs are 0.55 μm to 0.7 μm thick.
• Thermal isolation: The membrane hangs on thin legs or beams of silicon dioxide or silicon nitride, which have a low conductance; removing the air, or lowering its pressure, raises the responsivity (2.3 times in a device of 96 thermocouples).
• Responsivity and speed: The responsivity is proportional to the thermal resistance, whereas the time constant is the heat capacity times the same resistance, so that the ratio of responsivity to time constant depends only on the heat capacity.
• Absorber: Gold black (above 90% at 10 μm) and silicon nitride layers (64% for 1300 nm) are used; an absorptance below 1 lowers the responsivity in proportion.
• Arrays: Imagers of six thermocouples per pixel with a pixel of 160 μm to 190 μm have been made, and a 128×128 array with a pitch of 50 μm to 60 μm was proposed.
• Figures of merit: The Seebeck coefficient, the number of thermocouples, the thermal conductance, the responsivity, the time constant, the resistance, the NEP and the detectivity.

2. Working Principle

• Absorption and heating: The absorber receives the power $\eta P$, and the membrane rises above the substrate by $\Delta T=\eta P{R}_{th}$ for a constant signal.
• Thermoelectric conversion: Each pair of an n-type and a p-type leg gives ${\alpha }_{12}\Delta T$ between the junctions, and the $N$ pairs in series add; the resistance of the chain grows with $N$ as well.
• Heat paths: The heat leaves by the solid legs and the membrane, by the gas and by radiation; the solid part dominates in vacuum, and the gas conducts 57% of the heat in the air in the 96-thermocouple device.
• Modulated signals: The response falls as $1/\sqrt{1+(2\pi f\tau {)}^2}$, so a chopped source needs a frequency well below $1/(2\pi \tau )$ for the full response.
• Readout: The voltage is amplified (a factor of 100 in one imager), and the circuit may sit on the same chip, since the process is CMOS-compatible.

3. Applications

• Infrared imaging: Uncooled thermopile arrays for thermal imaging, with a time constant of 25 ms that is enough for 30 frames per second in one imager.
• Non-contact sensing: Single sensors for remote temperature measurement and for the detection of presence or of gas; these uses are background and are not in the sources read.
• Slow thermal imaging: Thermopile pixels switched between frames, for frame times of 3 s or more.

Thermoelectric Conversion and Responsivity

Name	Formula	Related Formulae	Variables/Constants
Seebeck voltage of the thermopile	$V_{out}=N{\alpha }_{12}\Delta T$	${\alpha }_{12}={\alpha }_p-{\alpha }_n$ $V_{single}={\alpha }_{12}\Delta T$	$V_{out}:open-circuitvoltage$ $N:numberofthermocouplesinseries$ $\begin{array}{r}{\alpha }_{12}:differenceoftheSeebeckcoefficients\\ ofthetwolegs(thesumoftheir\\ magnitudeswhenthesignsareopposite)\end{array}$ $\begin{array}{r}{\alpha }_p,{\alpha }_n:Seebeckcoefficientsofthep-type\\ andthen-typeleg\end{array}$ $\begin{array}{r}\Delta T:differenceofthetemperatureofthe\\ hotandthecoldjunctions\end{array}$ $\begin{array}{r}(N=96and{\alpha }_{12}=137+118=255\mu V/Kgive\\ 24.5mV/K;thesourcestates225\mu V/Kfor\\ thisdevice,computed)\end{array}$
Responsivity	$\Re =\eta N{\alpha }_{12}{R}_{th}=\frac{\eta N{\alpha }_{12}}{G}$	$\Delta T=\eta {\phi }_{0}A{R}_{th}$ $R_{th}=\frac{1}{G},{\Re }_{abs}=N{\alpha }_{12}{R}_{th}$	$\Re :responsivityperunitofincidentpower$ $\eta :absorptance$ $R_{th}:thermalresistancetothesubstrate$ $G:thermalconductance$ ${\phi }_0:incidentpowerdensity$ $A:areaoftheabsorber$ $\begin{array}{r}{\Re }_{abs}:responsivityperunitofabsorbed\\ power(thedefinitionofonesource)\end{array}$ $\begin{array}{r}(N=96,{\alpha }_{12}=255\mu V/K,\eta =1andG=0.12\\ mW/Kinvacuumgive204V/W,and0.28mW/K\\ inairgives87V/W,against202.8and\\ 88.5measured,computed)\end{array}$
Thermal conductance	$G=G_s+G_g+G_r$	$G_s=N{\sum }_i\frac{{\lambda }_i{w}_i{d}_i}{l_i}$ $G_r=4\xi {\sigma }_{SB}{A}_r{T}_0^3$	$\begin{array}{r}{G}_s:conductionthroughthesolidlegsand\\ membrane\end{array}$ $G_g:conductionofthegas$ $G_r:radiation$ $\begin{array}{r}{\lambda }_i,w_i,d_i,l_i:conductivity,width,\\ thicknessandlengthofthelayeri\end{array}$ $\xi :emissivity$ ${\sigma }_{SB}:Stefan-Boltzmannconstant$ $A_r:radiatingarea$ $\begin{array}{r}{T}_0:ambienttemperature,invacuumatlow\\ temperatureG\approx G_s\end{array}$ $\begin{array}{r}(poly-siliconlegsof183and198\mu \mathrm{m},5\mu \mathrm{m}\\ wideand0.55\mu \mathrm{m}thick,with96strips,\\ conduct90\mu W/Kandwith130and145\mu \mathrm{m}\\ and116strips152\mu W/K;airadds0.16\\ mW/Kto0.12,i.e.57\%,computed)\end{array}$
Time constant and speed of response	$\tau =C_{th}{R}_{th}=\frac{C_{th}}{G}$	$C_{th}={\sum }_i{\rho }_i{c}_i{V}_i$ $\frac{\Re }{\tau }=\frac{\eta N{\alpha }_{12}}{C_{th}}$	$\tau :thermaltimeconstant$ $C_{th}:heatcapacityofthelegsandtheabsorber$ $\begin{array}{r}{\rho }_i,c_i,V_i:density,specificheatand\\ volumeoftheparti\end{array}$ $\begin{array}{r}\Re /\tau :ratiothatdoesnotdependonG,soa\\ lowerconductancetradesspeedfor\\ responsivity\end{array}$ $\begin{array}{r}(2100V/Wand25msgive8.4\times 10^{4}V/Ws;\\ 1151V/Wand14.46msgive7.96\times 10^4,and\\ thetheoryofthesamedevice,256V/Wand\\ 13.09ms,gives1.95\times 10^4,computed)\end{array}$

Noise, Radiation Input and Response

Name	Formula	Related Formulae	Variables/Constants
Resistance and Johnson noise	$V_n=\sqrt{4k_{B}T{R}_0\Delta f}$	$R_0=N{\sum }_i\frac{{\gamma }_i{l}_i}{w_i{d}_i}$ $V_n\propto \sqrt{N}\left(R_0\propto N\right)$	$V_n:Johnsonnoisevoltage$ $k_B:Boltzmannconstant$ $T:temperature$ $R_0:resistanceofthethermopile$ $\Delta f:bandwidth$ $\begin{array}{r}{\gamma }_i:resistivityofthelayeri(thesum\\ wasreconstructedfromagarbledequation)\end{array}$ $N:numberofstrips$ $\begin{array}{r}(R_0=306k\Omega gives71nV/\sqrt{\mathrm{Hz}}and458.5k\Omega \\ gives87nV/\sqrt{\mathrm{Hz}}at300K,computed;the\\ sourcesgivenoothernoiseterm)\end{array}$
Noise-equivalent power and detectivity	$NEP=\frac{V_n}{\Re }$	$D^{\ast }=\frac{\sqrt{A\Delta f}}{NEP}$ $D^{\ast }=\Re \sqrt{\frac{A}{4k_{B}T{R}_0}}$	$NEP:noise-equivalentpower$ $V_n:noisevoltage$ $\Re :responsivity$ $D^{\ast }:specificdetectivity$ $A:areaoftheabsorber$ $\Delta f:bandwidth$ $\begin{array}{r}{R}_0:resistance,thelastformisforthe\\ bandwidthof1Hz\end{array}$ $k_B:Boltzmannconstant$ $\begin{array}{r}(R_0=306k\Omega ,\Re =88.5V/WandA=10^5\mu {\mathrm{m}}^2\\ giveNEP=0.80nW/\sqrt{\mathrm{Hz}}and{D}^{\ast }=3.9\times 10^7\\ \mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W(computed);thesourceprints\\ 0.81\times 10^{-6}Wand1.24\times 10^8)\end{array}$
Power received from a blackbody	$P_{abs}=\eta {\sigma }_{SB}\left(T_s^4-T_0^4\right){\sin }^2\theta A$	${\phi }_0={\sigma }_{SB}\left(T_s^4-T_0^4\right){\sin }^2\theta$ ${\sin }^2\theta =\frac{{\phi }_0}{{\sigma }_{SB}\left(T_s^4-T_0^4\right)}$	$P_{abs}:absorbedpower$ $\eta :absorptance$ $T_s:temperatureoftheblackbody$ $T_0:ambienttemperature$ $\begin{array}{r}\theta :half-angleoftheapertureofthe\\ sourceseenfromthedetector\end{array}$ $A:areaoftheabsorber$ ${\phi }_0:powerdensityatthedetector$ $\begin{array}{r}(500Kand295K(assumed)give3115W/{\mathrm{m}}^2\\ attheaperture;66.72W/{\mathrm{m}}^{2}at9cmmeans\\ {\sin }^2\theta =0.0214,\theta =8.4^{\circ }andanaperture\\ radiusof1.33cm,computed)\end{array}$
Frequency response and frame rate	$V\left(f\right)=\frac{V_0}{\sqrt{1+{\left(2\pi f\tau \right)}^2}}$	$f_{3dB}=\frac{1}{2\pi \tau }$ $\tau <T_{frame}$	$V\left(f\right):outputforasignalmodulatedatf$ $V_0:outputforaconstantsignal$ $\tau :thermaltimeconstant$ $\begin{array}{r}{f}_{3dB}:frequencyatwhichtheresponsefalls\\ to0.707\end{array}$ $T_{frame}:timeofaframe$ $\begin{array}{r}(achopperat4Hzand\tau =14.46msgive\\ 0.94oftheconstantresponse,and25ms\\ gives0.85;f_{3dB}=6.4Hzfor25ms;30fps\\ is33msperframe,computed)\end{array}$
Stress of the supporting film	$\sigma =\frac{E{h}^2}{6\left(1-\nu \right)R_{m}t}$	$R_m=\frac{R_a{R}_b}{R_b-R_a}$ $\frac{{\sigma }_{max}}{\sigma }=\frac{{\sigma }_f}{\sigma }$	$\sigma :residualstressofthefilm$ $E:Youngmodulusofthesubstrate$ $h:thicknessofthesubstrate$ $\nu :Poissonratio$ $\begin{array}{r}{R}_m:radiusofcurvaturefromtheradii\\ before(R_a)andafter(R_b)thedeposition\end{array}$ $t:thicknessofthefilm$ $\begin{array}{r}{\sigma }_f:fracturestrength,thesignconvention\\ for{R}_{m}isnotstatedbythesource\end{array}$ $\begin{array}{r}(240MPaagainstafracturestrengthof\\ 364MPaisamarginof1.5;asilicon\\ nitridefilmof1.32to1.54GPa(two\\ readingsofthesource)became0.55GPa\\ withanimplantofSi,i.e.36\%to42\%,\\ computed)\end{array}$

Important Questions

Ques: Is the Seebeck difference of 225 μV/K of the stacked device consistent with its responsivity?

No. The paper states ${\alpha }_{12}=225\mu V/K$ for the stacked-double-layer device, but its own coefficients give $118+137=255\mu V/K$ (and 240 from the other table). With $N=96$ and the conductances 0.12 and 0.28 mW/K, the responsivity is 180 and 77 V/W for 225, and 204 and 87 V/W for 255 (computed); the measured 202.8 and 88.5 V/W match 255. The ratio of the responsivities in vacuum and air, 2.29, matches the ratio of the conductances, 2.33 (computed). So 225 is probably a transposition of 255, which is my reading; the paper does not say, and the coplanar values 116.2 and 138.8 do follow from its aluminium value. Reference: Development of a thermopile infrared sensor using stacked double polycrystalline silicon layers based on the CMOS process, J. Micromech. Microeng. 23, 065026 (2013).

Ques: Are the NEP and the detectivity of that device consistent?

Not with a common bandwidth. The Johnson noise of 306 kΩ at 300 K is 71 nV/√Hz, which with 88.5 V/W gives an NEP of 0.80 nW/√Hz, and with $A=10^5\mu {\mathrm{m}}^2$ a detectivity of $3.9\times 10^7\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}/W$ (computed). The paper prints 0.81×10${}^{-6}$ W and $1.24\times 10^8$. The printed NEP is 1000 times mine, which is the square root of a bandwidth of $10^6$ Hz, and the printed detectivity is the one for $10^7$ Hz, for all four devices in vacuum and in air (inferred, computed). The paper does not state the bandwidth; it is an inference, and the same device has two bandwidths. Reference: Development of a thermopile infrared sensor using stacked double polycrystalline silicon layers based on the CMOS process, J. Micromech. Microeng. 23, 065026 (2013).

Ques: Is the thermal conductance of the stacked device consistent?

The text gives 69% and later 56% for the share of the air in the conductance; 0.28 and 0.12 mW/K give 57% (computed). The conductances per thermocouple are $0.1385\times 10^{-5}$ W/K for the stacked device, and with 96 thermocouples they sum to 0.133 mW/K (computed), more than the total of 0.12 mW/K in vacuum. Either the conductance per thermocouple includes a part that the total does not, such as the shared membrane, or the total or the number of thermocouples is not as printed; the paper does not say. The Seebeck coefficients and the conductivities also differ between its tables and its text (-110 and 130 against -118 and 137 μV/K). Reference: Development of a thermopile infrared sensor using stacked double polycrystalline silicon layers based on the CMOS process, J. Micromech. Microeng. 23, 065026 (2013).

Ques: What Seebeck coefficients do the sources use, and do they agree?

They do not. The stacked device is measured at -118 μV/K (n) and 137 μV/K (p) in the text and -110 and 130 in another table; the beam devices of 96 and 116 strips use -124.17 and 105.76 μV/K for doping of $3.64\times 10^{20}$ and $1.82\times 10^{20}\mathrm{c}{\mathrm{m}}^{-3}$; the imager of 2000 gives 0.24 mV/K for both types, without signs, for a dose of $10^{16}\mathrm{c}{\mathrm{m}}^{-2}$; and the optimization chapter uses 300 μV/K as an example. The aluminium leg is -1.66 or -1.8 μV/K. The spread is 106 to 240 μV/K for a leg, so a design needs the value for its own doping, and the source values cannot be pooled. Reference: Development of a thermopile infrared sensor using stacked double polycrystalline silicon layers based on the CMOS process, J. Micromech. Microeng. 23, 065026 (2013); Thermopile with double-ended and free-ended beams, hjlaser.henau.edu.cn (authors, title and year not shown); Hirota et al., Thermopile infrared imager, Sensors and Materials 12(5) (2000); Optimization of a thermopile thermo-sensitive element for an infrared sensor array, Springer chapter 10.1007/978-3-319-62870-7_5 (authors not shown).

Ques: Can the 2100 V/W and 25 ms of the imager of 2000 be checked?

Only with assumptions. With six pairs and ${\alpha }_{12}=0.48$ mV/K (opposite signs, my assumption), 2100 V/W means $R_{th}\eta =7.3\times 10^5$ K/W, i.e. $G=1.4\mu W/K$ for $\eta =1$ and, with 25 ms, $C_{th}=3.4\times 10^{-8}$ J/K (computed). If the 100-fold amplifier is included in the 2100 V/W, which the paper does not say, the true response is 21 V/W and $G=137\mu W/K$ (computed). Neither $G$ nor $C_{th}$ is given. The values hold at 7.33 Pa; no value in air is given. The paper’s remark that 25 ms is enough for 30 fps is correct on its face, as 33 ms is the frame time, and the margin is small. Reference: Hirota et al., Thermopile infrared imager, Sensors and Materials 12(5) (2000).

Ques: Why does the measured responsivity of the beam devices exceed the theory by 4.5 times?

The theory gives 255.83 V/W against 1151.14 measured (4.50) for the double-ended device and 148.89 against 656.23 (4.41) for the free-ended one (computed). The poly-silicon legs alone conduct 90 and 152 μW/K (computed), which with 96 and 116 thermocouples and 230 μV/K gives 244 and 176 V/W, close to the theory, so the theory is plausible and the measurement is the outlier. The paper’s text gives a signal of 7.47 mV, its table 7.34 V, and 1151 V/W with the power of 6.67 μW gives 7.68 mV (computed). The 66.72 W/m${}^2$ at 9 cm implies an aperture radius of 1.33 cm (computed). The gap is not explained. Reference: Thermopile with double-ended and free-ended beams, hjlaser.henau.edu.cn (authors, title and year not shown).

Ques: What does the ratio of responsivity to time constant tell about the devices?

With $\tau =C_{th}/G$ and $\Re =\eta N{\alpha }_{12}/G$, the ratio $\Re /\tau =\eta N{\alpha }_{12}/C_{th}$ does not depend on the conductance (computed from the formulas). It is $8.4\times 10^4$ V/Ws for the imager of 2000, $7.96\times 10^4$ for the measured double-ended beam device and $5.6\times 10^4$ for the free-ended one, but only $1.95\times 10^4$ and $2.1\times 10^4$ for their theory (computed). So a four-fold larger measured response comes with an unchanged time constant, which means that the heat capacity would be four times smaller than in the theory, or the response is overstated. The paper does not give the heat capacity, and because the conductance cancels, the ratio is a clean test of any heat capacity that a paper states. Reference: Thermopile with double-ended and free-ended beams, hjlaser.henau.edu.cn (authors, title and year not shown); Hirota et al., Thermopile infrared imager, Sensors and Materials 12(5) (2000).

Ques: How much does the absorber matter?

The responsivity per incident power is proportional to $\eta$. The stacked device is quoted for $\eta =1$ (202.8 V/W in vacuum) and 162.3 V/W for $\eta =0.8$, which is 0.8 times the former (computed). This scaling is for the incident power; with the definition of the paper, $V/P_{absorb}$, the absorptance would cancel, so the definition and the table disagree. Gold black is quoted above 90% at 10 μm, and a silicon nitride layer of 1300 nm gives 64% in the optimization chapter. The absorber also adds heat capacity, so a thicker layer lowers the speed; an absorptance of 0.9 would give 183 V/W in vacuum for this device (computed). Reference: Development of a thermopile infrared sensor using stacked double polycrystalline silicon layers based on the CMOS process, J. Micromech. Microeng. 23, 065026 (2013); Hirota et al., Thermopile infrared imager, Sensors and Materials 12(5) (2000); Optimization of a thermopile thermo-sensitive element for an infrared sensor array, Springer chapter 10.1007/978-3-319-62870-7_5 (authors not shown).

Ques: What happens in vacuum, and how large is the stress margin of the membrane?

Removing the air raises the responsivity by 2.29 and, since $\tau =C/G$, the time constant by about the same factor (2.33 for the conductances) if the heat capacity is unchanged (computed); the stacked device gives no time constant, and the imager gives 25 ms at 7.33 Pa and no value at atmospheric pressure. The stress of the membrane of the stacked device is 240 MPa against a fracture strength of 364 ± 57 MPa, a margin of 1.5 (computed); a nitride film of 1.32 to 1.54 GPa (two readings of the source) fell to 0.55 GPa with a silicon implant. The paper’s expression for the mean radius has a sign that is not stated, so the stress formula must be checked. Reference: Development of a thermopile infrared sensor using stacked double polycrystalline silicon layers based on the CMOS process, J. Micromech. Microeng. 23, 065026 (2013); Hirota et al., Thermopile infrared imager, Sensors and Materials 12(5) (2000).

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