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Fakultät Elektrotechnik und Medientechnik

Selected Topics in Micro- and Nanoelectronics Practical Training Attempt: 3 ω Omega Measurement

Group:

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Date:

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Name:

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A well done homework is the requirement to participate in this practical training. Without preparation the training will be classified as failed and it must take place a replacement term. Only one replacement term is available for one group. Otherwise there is no admission to write the examination. At the beginning of your training a short colloquium will be held in which the understanding of the preparation is checked.

The preparation was taken from a student work, therefore some references to chapters are included, which are not part of this manual.

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Content: 1

2

3

Introduction....................................................................................................................................................... 3 1.1

List of Symbols ........................................................................................................................................... 3

1.2

Heat conduction.......................................................................................................................................... 4

1.3

Silicon dioxide............................................................................................................................................. 5

1.4

Heater and Sample Structure ..................................................................................................................... 5

1.5

Design Rules for Samples .......................................................................................................................... 9

1.6

Theoretical Background of 3 Omega Method ............................................................................................11

1.7

Experimental Set-up of 3 Omega Method ................................................................................................ 14

1.8

Theory of performing a 3 ω Omega Measurement ................................................................................... 15

1.9

Accuracy of measurement equipment ...................................................................................................... 20

Homework ...................................................................................................................................................... 20 2.1

Preparation of Excel evaluation sheet for thermal conductivity λf ............................................................. 20

2.2

Preparation of Excel evaluation sheet for temperature coefficient of resistance α ................................... 22

2.3

Preparation Questions .............................................................................................................................. 23

Test ................................................................................................................................................................. 27 3.1

Practical performing a 3 ω Omega Measurement .................................................................................... 30

3.2

Measuring Alpha ....................................................................................................................................... 35

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1 Introduction 1.1 List of Symbols

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1.2 Heat conduction

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1.3 Silicon dioxide

Table 1: literature values of thermal conductivity for silicon dioxide

1.4 Heater and Sample Structure

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Figure 2/3

Figure 1: Schematic cross section of a sample showing the three layered heater with titanium at the bottom, platinum in the middle and gold on top.

Figure 2: Image of a whole wafer on the left. It can be seen, that the heaters are repeatedly applied in a straight line over the whole wafer. The wafer holds eight lines with a 8.5 mm distance between them and a width of 16.3 mm. The two images in the middle show a cross section view of the wafer. The upper one shows the cross section of the whole wafer, the lower one only shows the upper part including the thin film and parts of the heater structure. The images on the right give a better view on the heater structures itself, where the top one shows all 16 different heaters and the bottom one shows both contact pads on the right side of a single heater.

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Fakultät Elektrotechnik und Medientechnik 2,0 um 3,5 um 5,0 um 6,0 um 2,0 um 3,5 um 5,0 um 6,0 um …….

9 mm 13 mm 15 mm 11 mm

Figure 3: Set of 16 different heaters captured using the LSM and an optical stitching technique. It can be seen, that the lengths are near the stated values.

Figure 1/4

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Figure 4: Three different measurements of a heater cross section, all showing, that the heater is no perfect rectangle. The sides have a noticeable slope. First measurement was done using a LSM; second measurement was done using an AFM in contact mode; last measurement is a simple cross section embedded in resin, polished and then observed using an optical microscope.

Table 2: comparing the heater width of a 9 mm long heater for different film thicknesses

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1.5 Design Rules for Samples

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1.6 Theoretical Background of 3 Omega Method

Figure 5: 3ω metal heater deposited on top of a SiO 2 thin film with length l, width 2b and two contact pads on each side. Heaters in this work have theoretical length of 9 mm to 15 mm and width of 2 µm to 6 µm. The image is not to scale.

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2

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1.7 Experimental Set-up of 3 Omega Method Figure 6, Figure 7)

Figure 6: Bridge circuit used in experimental set-up. Ri is the internal resistance of the LIA, R1 and R2 are fixed high precision resistances, Rv is the decade resistance and Rs is the sample resistance. The circles show the position of the different plugs.

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1.8 Theory of performing a 3 ω Omega Measurement

Figure 7: Complete experimental set-up. An oscilloscope, a lock-in amplifier, a decade resistance, a DC power supply, the bridge circuit, the control box for the peltier element, a optical microscope, the sample, the contact needles, a peltier element and the vacuum globe can be seen.

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Figure 8.

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Figure 8: Lock-In control elements on the left. The horizontal buttons allow for switching between different menu points. The vertical buttons allow interaction with the blue side bar of each menu. The knob and the four buttons below are for further navigation. The Lock-In menu shows us the real and imaginary part of our signal and the bridge voltage from top to bottom. The Ref option on the side bar allows for setting balancing frequency and voltage. The Spectra menu shows the current measured data. Here measurements can also be started and data saved. The Acquire option allows for setting start and ending frequency, total measurement points and measurement delay.

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Figure 9: Temperature dependent resistance of a 13 mm long, 2 µm wide and 580 nm high heater structure on a 1 µm SiO2 film.

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Fakultät Elektrotechnik und Medientechnik Meaurement Evalutaion sheet

Figure 10: Excel sheet for calculating the thermal conductivity using the measurement results

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1.9 Accuracy of measurement equipment

Table 3: Different measurement errors for all measured variables calculated for a SiO 2 sample with film thickness of 100nm and heater length and width of 11mm and 6µm, respectively. The voltage U(g 1 ) used was 7 V. The different relative errors for the multimeter occur due to a single measurement for R s and multiple different measurements for α R decreasing the error.

2 Homework 2.1 Preparation

of

Excel

evaluation

sheet

for

thermal

conductivity λf On ilearn you will find the empty template of the excel sheet for the evaluation of the measurement. Make yourself familiar with it. The dependencies of the formulas are shown either in the test part in chapter 3 and in the excel sheet itself. Enter the correct formulas in excel for the calculations. With the given values you can check the correctness of your formulas using the results on these images.

 Take over the cells in column D in excel that have not yet been filled in from the template on the next page.

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 Formula for Heating power generated by AC heating current in Cell D24  Hint for the first step: =D21^2/D11

Figure 11: Excel sheet for calculating the thermal conductivity – Part 1

 Formula for 3w Voltage in complete Column J (A fixed cell reference for example cell D15 is created with $D$15.)

Figure 12: Excel sheet for calculating the thermal conductivity – Part 2

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 Formulas for complete Columns L, M, N and P

Figure 13: Excel sheet for calculating the thermal conductivity – Part 3

 Formula for Mean Value of Thermal film conductivity in Cell R11. (Use just the mean value of the linear range of Column P. Below in the sheet you find a diagram of thermal conductivity to find out the linear range, red marked)

Figure 14: Diagram thermal conductivity λf over frequency

2.2 Preparation of Excel evaluation sheet for temperature coefficient of resistance α In order to generate a good detectable 3ω voltage in the metal heater, the heater material must have a high temperature coefficient of resistance α. The heater material in our experiment will be gold. For calculating the temperature oscillation ∆𝑇 out of the 3ω voltage the exact temperature coefficient of resistance of the golden heater will be necessary. Therefore, α is determined before 3ω method is applied. Practical Training – 3ω Measurement v.13

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For this purpose, the electrical resistance of the metal heater is measured by a multimeter at several temperatures. Then, a fitting line is drawn through the measurement points, by which the temperature coefficient of resistance con be calculated. Prepare an Excel sheet for evaluation of the temperature coefficient of resistance which look like the sheet illustrated in following figure.

Figure 15: Excel sheet for calculating temperature coefficient of resistance α

2.3 Preparation Questions a) Why is this measurement called 3w method? ………………………………………………………………………………………………… b) Which other methods do you know to measure the thermal conductivity? ………………………………………………………………………………………………… ………………………………………………………………………………………………… c) What is the reason in this experiment for using a: Lock-In Amplifire:

……………………………………………………………………

Oscilloscope:

……………………………………………………………………

Resistor Decade:

……………………………………………………………………

R1 and R2:

……………………………………………………………………

Vacuum globe:

……………………………………………………………………

Peltier element:

……………………………………………………………………

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d) Explain shortly the Mechanisms of heat conduction in gases, liquids, conductive solids, non-conductive solids. ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… e) Write the simple heat conduction equation (parallel surfaces)? ………………………………………………………………………………………………… ………………………………………………………………………………………………… f) Why do we need two inputs in our lockin-amplifier? ………………………………………………………………………………………………… ………………………………………………………………………………………………… g) Why is it important, that the resistance of Rv is in the same range as Rs (Rsample) ………………………………………………………………………………………………… h) Which thermal conductivity do we want to measure, in-plane or cross-plane? ………………………………………………………………………………………………… i) Write analogous (no formulas) all nine approximations of the design rules which are laid out to achieve a result with a theoretical error of less than 1 %. ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Practical Training – 3ω Measurement v.13

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………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… j) Sketch a diagram of real part of 3w Signal over frequency and mark the interesting frequency range for our evaluation.

Figure 16: Bridge circuit used in experimental set-up with values. Ri is the internal resistance of the LIA, R1 and R2 are fixed high precision resistances, Rv is the decade resistance and Rs is the sample resistance Practical Training – 3ω Measurement v.13

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k) Determine the value of RV for a measured value of Rs=130 Ohm in case of a balanced bridge (RMS means root mean square and therefore effective value). ………………………………………………………………………………………………… l) Calculate the voltage Ug2 for a source voltage of Ug1=5V.

m) Calculate the voltage UR2 = URs as a function of Ug2 in case of a balanced bridge. First calculate the formula and later the exact value.

n) Calculate the current I1 as a function of R1, R2 and Ug2

o) Calculate the current IV as a function of RS, RV and Ug2

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3 Test

Figure 17: Schematic Experimental Setup

Understanding the Excel Sheet: Ph (Heating power amplitude generated by AC heating current) Dependency: -

Voltage Wheatstone bridge right: URs Resistance Sample RS

𝑃ℎ =

𝑈(𝑅𝑠)² 𝑅𝑠

(heaterspecific) (heaterspecific)

𝑃ℎ = 𝐼(𝑣) ∗ 𝑅𝑠

U3ω (3ω Voltage) Dependency: -

Resistance Decade RV Resistance Sample RS Lock-In Voltage output W3ω 𝑈3𝜔 =

(heaterspecific) (heaterspecific) (present Lock-In value for this f)

𝑅𝑣 + 𝑅𝑠 ∗ 𝑊3𝜔 𝑅𝑣

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Temperature oscillation amplitude substrate Silicon ΔTs Dependency: - Thermal conductivity Silicon λs (material constant) - Specific heat capacity Silicon cp (material constant) - Density of Silicon ρ (material constant) - Ftting constant η (material constant) - Heater length l (heaterspecific) - Heater width 2b (heaterspecific) - Frequency f (present Lock-In value) - Amplitude of heating power Ph ∆𝑇𝑠 =

𝑃ℎ 𝜆𝑠 ( ) ∗ 0,5 ∗ ln    + 𝜂 − 0,5 ∗ ln 4 ∗ 𝜋 ∗ 𝑓  𝑙 ∗ 𝜋 ∗ 𝜆𝑠 2𝑏 𝑐𝑝 ∗ 𝜌 ∗ 󰇡 󰇢 2

Temperature coefficient of resistance α Dependency: - ΔR = change in resistance (Ω) - Ro = initial resistance (Ω) - ΔT = change in temperature (K) 𝛼=

𝛥𝑅 (𝑅𝑜𝛥𝑇)

Temperature oscillation amplitude substrate and film ΔTs + ΔTf Dependency: - Temperature coefficient of resistance α (constant a sample) - Voltage Wheatstone bridge left: UR2 (heaterspecific) - 3ω Voltage: U3ω ∆𝑇𝑠 + ∆ 𝑇𝑓 =

2 ∗ 𝑈3𝜔 𝛼 ∗ 𝑈( 𝑅2)

Temperature oscillation amplitude film ΔTf ∆𝑇𝑓 = (∆𝑇𝑓 + ∆𝑇𝑠) - ∆𝑇𝑠

Thermal conductivity film λf Dependency: - Heater thickness df (heaterspecific) - Heater length l (heaterspecific) - Heater width 2b (heaterspecific) - Heating power Ph generated by AC heating current - Temperature oscillation amplitude ΔTf 𝜆𝑓 =

𝑃ℎ ∗ 𝑑𝑓 2𝑏 ∗ 𝑙 ∗ ∆𝑇𝑓

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3.1 Practical performing a 3 ω Omega Measurement

Figure 18: Our Testsample

We want to perform a 3 ω Omega Measurement for a heater structure. Choose one structure as you like. For an overview look at figure 3! -

thickness 1µm SiO2 on Si Bulk

-

length l =

………………

-

width 2b =

………………

-

heather thickness df = 578nm alpha = 0,0030 1/K supply source voltage U(g1) = 5Vrms frequency range: 10 Hz to 10 kHz interesting f range: 50 Hz and 1 kHz

Substrat Parameters

Silicon

Thermal conductivity Silicon

λs

[W/(m*K)]

147

Specific heat capacity Silicon

cp

[J/ (kg*K)]

711

Density of Silicon

ρ

[kg/ (m3)] 2330

Ftting constant