Chapter 2: Sample Preparation and Experimental Setup
2.1 Suspended, Micro-Fabricated Thermal Bridge Approach
2.1.3 Measurement Setup
A schematic diagram of the experimental setup for thermal conductance measurements is given by Figure 2-4. Prior to measurement, the micro-fabricated thermal bridge device containing the sample of interest and the corresponding reference device are mounted on top of a ceramic dual in-line package (DIP) and placed in a variable temperature cryostat (Janis CCS-450). The vacuum level is then lowered to be at or below 10-6 Torr using a turbomolecular pump (Edwards E2M1.5) and the local temperature at the thermal measurement microdevice is adjusted via a temperature controller (Lakeshore 335). Lastly, the heating, sensing, and reference membranes are connected to two lock-in amplifiers (Stanford Research SR 850) which are used to monitor the voltage change of the PRTs on the relevant membranes.
During thermal measurement, a small sinusoidal alternating current (AC) signal, iac, is generated from the heating side lock-in amplifier and coupled to a direct current (DC) heating source (iac << I) by an integrated differential amplifier (Analog Devices SSM2141). To ensure a constant current condition under each specified DC heating voltage, the coupled current sources are connected in series to a resistor with a magnitude (500 KΩ) significantly larger than that of the
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individual heater side PRT, Rh (~10 KΩ). The Joule heat generated from the applied DC voltages is able to pass through the sample bridging the suspended membranes, and the respective temperature increases induce resistance changes in the embedded heating and sensing side PRTs (Rhand Rs) which can be measured via the 4-point probe method by lock-in amplifiers. This change in the output AC voltages of the lock-in amplifier is monitored by a data acquisition (DAQ) board (National Instruments PCI-6052e), and the DC heating voltage is simultaneously measured by a high accuracy current preamplifier (DL Instruments Model 1211).
Figure 2-4: Schematic representation of the common mode measurement setup.
To implement the Wheatstone bridge circuit into the measurement scheme, the sensing side thermometer, Rs, is connected to a PRT, Rs,ref, on a nearby (<5 mm) reference membrane. The close proximity of the measurement and reference membranes helped to minimize variations in local
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ambient temperature. The Wheatstone bridge circuit is then completed by connecting Rs and Rs,ref
to two precision resistors (Extech 3804400) which remain at room temperature ambient conditions.
Figure 2-5: Schematic representation of the thermal circuit of the common mode rejection measurement scheme.
A schematic of the thermal circuit of the common mode rejection measurement scheme is given in Figure 2-5. When a steady-state, DC current, I, is passed through the heating circuit, a certain amount of Joule heat is produced in the heating side PRT (Qh = I2Rh) and the platinum leads patterned on the support beams (2QL = 2I2RL), where Rh and RL are the electrical resistances of the PRT and the platinum leads, respectively. A portion of this heat, Q2, passes through the sample bridging the suspended membranes and raises the temperature of the sensing side membrane. Finite element analysis has previously shown that, because the thermal resistance of the suspended membranes is significantly smaller than that of the narrow support beams connecting them to the substrate at temperature T0, the heating and sensing side membranes approach a uniform temperature of Thand Ts, respectively (Moore & Shi 2010). In the high vacuum condition of the measurement environment and by ensuring the temperature rise of the heating
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membrane is small (∆Th = Th – T0 < 5 K), the background heat transfer between the suspended membranes due to convection and radiation can be considered negligible in comparison to the sample thermal conductance, Gs, as is discussed in detail below. Thus, in total, the heat transferred to the silicon substrate through the six support beams attached to the heating side membrane is calculated as Q1 = Qh + 2QL - Q2.
The beams which support the suspended membrane are designed to be identical and are fabricated simultaneously to minimize variations. If the radiation and convection heat transfer between the sensing side suspended membrane and the silicon substrate are taken to be negligible compared to the conduction heat transfer along the support beams, the total thermal conductance of the six beams can be written as Gb = 6kbA/L, where Gb is the thermal conductance, kb is the beam thermal conductivity, A is the cross-sectional area, and L is the length. By applying this to the thermal circuit shown in Figure 2-5, the beam thermal conductance can be related to the sample thermal conductance according to Eqn (2.1):
𝑄2 = 𝐺𝑏Δ𝑇𝑠 = 𝐺𝑠(𝑇ℎ− 𝑇𝑠), (2.1) where ∆Ts = Ts – T0 is the relative temperature rise of the sensing side membrane. The total measured thermal resistance of the sample, Rsam, is the inverse of the sample thermal conductance, Gs, and is the sum total of the intrinsic thermal resistance of the sample, Ri, and the contact thermal resistances that exists between the sample and the Pt electrodes, Rc, as given by Eqn. (2.2):
𝑅𝑠𝑎𝑚 = 𝑅𝑖+ 𝑅𝑐
.
(2.2)Here the intrinsic thermal resistance can be written as 𝑅𝑖 = 𝐿𝑛
𝜅𝑛𝐴𝑛, where Ln, κn, and An are the suspended length, intrinsic thermal conductivity, and the cross-sectional area of the sample bridging the two membranes, respectively. Because the temperature rise of the heating side
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suspended membrane, ∆Th, is controlled to be small (~5 K), the thermal conductance Gs, Gb and Gc are taken to be constant during the Joule heating process.
Further, the thermal conductance of the beam and sample can be expressed as functions of the total Joule heat generated in the Pt heaters and the relative temperature rises of the heating and sensing side suspended membranes such that:
𝐺𝑏 = 𝑄ℎ+ 𝑄𝐿
Δ𝑇ℎ+ Δ𝑇𝑠, (2.3)
and
𝐺𝑠 = 𝐺𝑏 Δ𝑇𝑠
Δ𝑇ℎ− Δ𝑇𝑠. (2.4)
Qh and QL can be calculated from the current and voltage drops across the heating side PRT and the two Pt leads, and ∆Th and ∆Ts are derived from the measured electrical resistance changes of the heating and sensing side PRTs and the temperature coefficient of resistance (TCR) of the platinum where TCR = (dR/dT)/R.
A simplified schematic of the electrical circuits used to measure changes in the heating and sensing side electrical resistances is shown in Figure 2-6. For the heating side temperature rise,
∆Th, changes in the electrical resistance are monitored via the four-point method by using a sinusoidal current of ~300 nA and 1400 Hz. A lock-in amplifier is used to monitor the first harmonic component of the voltage drop across the heating membrane PRT (vac), yielding 𝑅ℎ =
𝑣𝑎𝑐
𝑖𝑎𝑐. The corresponding temperature rise of the heating membrane can then be determined by monitoring the resistance changes of the heating PRT in response to a sweeping DC heating current as given by Eqn. (2.5):
37 Δ𝑇ℎ(𝐼)= 𝑅ℎ(𝐼)− 𝑅(0)
𝑑𝑅ℎ 𝑑𝑇
. (2.5)
Figure 2-6: A simplified schematic of the electrical measurement circuit.
Rather than measuring the resistance change directly, the temperature rise of the sensing membrane, ∆Ts, is obtained from the voltage changes of the Wheatstone bridge circuit. The bridge circuit consists of electrical resistance of the sensing side membrane, Rs, the electrical resistance of the reference membrane, Rs,ref, and the electrical resistance of two additional precision resistors, R2 and R3, which are connected in parallel. The sensing side lock-in amplifier supplies the source voltage, vs, and simultaneously measures the bridge output voltage, vg, with this latter term being defined as the difference between the two branches, vA and vB. The relationship between the source and output voltages is given by Eqn. (2.6):
v𝑔 = v𝐴 − v𝐵 = ( 𝑅2
𝑅𝑠+𝑅2− 𝑅3
𝑅𝑠,𝑟𝑒𝑓+𝑅3) v𝑠. (2.6)
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By rearranging the above equation, the electrical resistance of the sensing side can be expressed as:
𝑅𝑠 = 𝑅2 ( 𝑅3
𝑅𝑠,𝑟𝑒𝑓+𝑅3+𝑣𝑔 v𝑠)
− 𝑅2. (2.7)
However, unlike the heating side resistance, the electrical resistance of the sensing side includes the resistance of the sensing PRT, Rc, as well as the resistance of two Pt leads on the support beams, 2RL. Additionally, while the temperature is uniform across the sensing membrane, it varies linearly along the support beams from Ts to T0. Owing to this, the sensing side total electrical resistance is then:
𝑅𝑠+ Δ𝑅𝑠 = 2 [𝑅𝐿+𝑑𝑅𝐿
𝑑𝑇 (𝑇𝑠− 𝑇0
2 )] + 𝑅𝑐 +𝑑𝑅𝑐
𝑑𝑇 (𝑇𝑠− 𝑇0), (2.8) where ∆Rs is the temperature-induced resistance change of the sensing side and can be expressed according to Eqn. (2.9):
Δ𝑅𝑠 = (𝑑𝑅𝐿
𝑑𝑇 +𝑑𝑅𝑐
𝑑𝑇) (𝑇𝑠− 𝑇0). (2.9)
The resistance change of the sensing side can then be regarded as the result of a uniform temperature change across the sensing side membrane and a single support beam. Therefore, an effective resistance change per temperature can be defined as:
𝑑𝑅𝑠,𝑒𝑓𝑓
𝑑𝑇 = ( 𝑅𝐿+ 𝑅𝑐 2𝑅𝐿+ 𝑅𝑐)𝑑𝑅𝑠
𝑑𝑇. (2.10)
Finally, the temperature rise of the sensing side is able to be calculated according to Eqn.
(2.11):
39 Δ𝑇𝑠 = Δ𝑅𝑠
𝑑𝑅𝑠,𝑒𝑓𝑓 𝑑𝑇
. (2.11)
Thus, the results of Eqn. (2.5) and Eqn. (2.11) can be used to calculate the total thermal conductance of the suspended sample according to Eqn. (2.3) and Eqn. (2.4).