Full thermoelectric characterization of InAs nanowires using MEMS heater/sensors

The relevant fabrication steps of the MEMS heaters/sensors are depicted in figure 2. We started with silicon (111) wafers (figure 2(a)) covered with a 150 nm thick ultra-low-stress silicon nitride (SiN) layer (obtained from SiMat). Contact pads, consisting of 50 nm Au on 2 nm Cr, were fabricated via a bi-layer lift-off process using the resists LOR5B and AZ6612), sputter deposion, and lift-off in N-Methyl-2-pyrrolidon. The electrical heaters (figure 2(b)) were defined by electron-beam lithography (Vistec EBPG5200, and Raith e-Line) of a 340 nm thick bi-layer resist (methylmethacrylate and polymethylmethacrylate), ${{O}_{2}}$ plasma cleaning, thermal metal deposition (2 nm Cr, 50 nm Au), lift-off in acetone, rinsing with isopropanol, and ${{O}_{2}}$ plasma cleaning.

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Another optical lithography step with AZ6612 resist was carried out to selectively remove SiN using reactive ion etching (figures 2(c)–(e)). The SiN was under-etched using wet or dry etching (figure 2(f)). For wet etching, we used tetramethylammonium hydroxide (25% concentration for 30 min at 80$\;{}^\circ$C) and subsequent replacement of solvent using isopropanol, which is then removed in a critical point dryer. Similar results were achieved using dry etching (deep reactive ion etching in ${\rm S}{{{\rm F}}_{6}}$ plasma in an Alcatel AMS200 tool). An example of a fabricated heater/sensor device is shown in figures 1(a), (b).

InAs NWs were grown using selective area epitaxy in a metal-organic chemical-vapor-deposition tool (Veeco). Si wafers with (111) orientation were pretreated by first dipping them into buffered hydrofluoric acid, followed by oxidation in air at 100 ${}^\circ$C for 20 min to form an oxide mask with pinholes. NW growth was carried out at 520 ${}^\circ$C for 90 min using a tertiary-butyl-arsine flow of 20 μmol ${\rm mi}{{{\rm n}}^{-1}}$ and a trimethyl-indium flow of 1.15 μmol ${\rm mi}{{{\rm n}}^{-1}}$. Under these conditions, InAs NWs grow perpendicular to the Si substrate along the $\left\langle 111 \right\rangle$ direction with predominant zincblende crystal structure [36]. The resulting InAs NWs had a diameter $d\sim 120$ nm and a total length of 4–5 μm. A transmission electron micrograph of one of these NWs is shown in figure 1(a). The NWs have a hexagonal cross-section with very low surface roughness. However, the crystal structure is affected by a multitude of stacking faults of the crystal planes perpendicular to the growth direction.

Figure 3 also shows images taken between different steps of the preparation procedure. The NW samples were positioned and contacted to the heater/sensor devices using a 3D-micro manipulator and a tungsten needle tip inside a combined electron microscope/focused ion beam (FIB) tool (FEI). Starting point was an under-etched MEMS heater/sensor structure as shown in figure 1(b). NWs were transferred onto a silicon oxide or SiN surface by tissue sweeping or stamping with polydimethylsiloxane (PDMS). Immediately after this transfer step the manipulation was started, because otherwise chemical binding forces or oxidation may lead to an increased adhesion of NWs to substrate which makes manipulation difficult. Individual NWs were picked up through adhesive forces using a tungsten needle, which was controlled using micromanipulators (Omniprobe). Typically the adhesion forces through van-der-Waals or electrostatic attraction is sufficient to adhere the NW to the tungsten needle. To influence the strength of adhesion between the needle and the NW, we used an electrically floating needle (negatively charged through the electron beam) during NW pick-up, and an uncharged needle (electrically grounded) while releasing the NW. During release of the NW onto the under-etched membranes the wire ends were touching the predefined leads near each heater/sensor, as shown in figure 3(b). At this stage, the two MEMS heaters/sensors were still connected through two bridges in the SiN membrane to provide additional stability during the NW placement (figure 3(b)).

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Then, an electrically conductive contact was formed between the NW and the predifined leads (figures 3(c) and (d)). For this, the contact regions of the NW were first exposed to a well-controlled gallium-ion beam, such that only the native oxide layer of the NW was locally removed. Otherwise, care was taken to avoid ion-beam exposure of any other areas of the NW. To this end, all imaging steps were performed only with the electron beam of the dual-beam FIB tool. The contact regions were then metalized using ion-beam-induced deposition of platinum from an organo-platinum precursor (methylcyclopentadienyl (trimethyl) platinum). NW segments that extended too far into the heater region, were cut off using the focused gallium-ion beam (also shown in figure 3(c)). Finally, the SiN membrane bridges were cut using gallium-ion beam milling such that the MEMS heaters/sensors were connected to each other only via the NW. An example of a final device ready to be measured is shown in figure 3(d).

The thermal transport measurements using MEMS heaters/sensors rely on Joule heating and the temperature coefficient of resistance (TCR) to provide and measure a temperature gradient across the NW. The electrical four-probe resistance of the heaters was determined from current-voltage characteristics. For temperature calibration, the resistance in the limit of zero current was used. In this limit, it is safe to assume that the heaters/sensors have the temperature of the chip, which was measured independently. The TCR was determined for each heater separately by varying the chip temperature with the cryostat. It is assumed that Joule heating provides uniform temperature distribution on the heater platform.

In a typical measurement, the temperature of one heater/sensor (${{T}_{H1}}$) is raised by applying a heating current (${{I}_{1}}$) while its electrical resistances (${{R}_{H1}}$) is measured. Simultaneously, the temperature of the second heater/sensor (${{T}_{H2}}$) is determined by measuring its electrical resistance (${{R}_{H2}}$) using a small probing current (${{I}_{2}}$). To determine the thermal conductance of the NW sample, ${{G}_{S}}$, from the measurement, both the heater temperatures, ${{T}_{H1}}$ and ${{T}_{H2}}$, and the heat flux through the NW sample, ${{Q}_{S}}$, have to be determined (see figure 1 (d)). The heat dissipated in the heaters, ${{Q}_{JH1}}$ and ${{Q}_{JH2}}$, can be calculated from the respective heater currents and the electrical four-probe resistances. The two-probe resistance measured simultaneously allows us to determine the Joule dissipation in the electrical leads connecting to the heaters, ${{Q}_{L1}}$ and ${{Q}_{L2}}$. These electrical measurements of Joule dissipation and heater temperatures are related to the heat flux through the NW sample and thereby to the thermal conductance using the analytical treatment as described below.

The aspect ratio of the SiN platforms connecting the heaters to the carrier chip are so large that the application of the one-dimensional heat-diffusion equation is justified. In steady state, the heat-diffusion equation along the legs leading to heater 1 becomes,

$$\begin{eqnarray}&&\frac{{{Q}_{JL1}}}{L}=-\kappa A\frac{{{{\rm d}}^{2}}\Delta T}{{\rm d}{{x}^{2}}},\end{eqnarray} \tag{ 1 }$$

expressing the temperature distribution along the legs as the temperature rise $\Delta T\left( x \right)$ above ambient (or chip) temperature as a function of position x along the beam of length L. Here, ${{Q}_{JL1}}$ is the Joule dissipation along the legs. The cross-sectional area A and the thermal conductivity κ comprise both SiN and metal leads and are therefore only effective numbers in this calculation that defines the thermal conductance of the legs, ${{G}_{L}}=\kappa A/L$.

With the boundary conditions

$$\begin{eqnarray}&&\Delta T\left( x=0 \right)=0\quad \Delta T\left( x=L \right)=\Delta {{T}_{H1}},\end{eqnarray} \tag{ 2 }$$

(1) can be solved using

$$\begin{eqnarray}&&\Delta T\left( x \right)=-\frac{{{Q}_{JL1}}}{2L\kappa A}{{x}^{2}}+\left( \frac{\Delta {{T}_{H1}}}{L}+\frac{{{Q}_{JL1}}}{2\kappa A} \right)x.\end{eqnarray} \tag{ 3 }$$

From this, we determine the total heat flow from the legs to the ambient for heater 1

$$\begin{eqnarray}&&{{Q}_{L1}}=\kappa A{{\left. \frac{\partial \Delta T}{\partial x} \right|}_{x=0}}={{G}_{L1}}\Delta {{T}_{H1}}+\frac{{{Q}_{JL1}}}{2}\end{eqnarray} \tag{ 4 }$$

and for the second heater accordingly. Next, we invoke energy conservation:

$$\begin{eqnarray}&&{{Q}_{L1}}={{Q}_{JH1}}+{{Q}_{JL1}}-{{Q}_{S}}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{{Q}_{L2}}={{Q}_{JH2}}+{{Q}_{JL2}}+{{Q}_{S}}\end{eqnarray} \tag{ 6 }$$

where we introduced the Joule heat dissipated in the heaters, ${{Q}_{JH1}}$ and ${{Q}_{JH2}}$, and along the NW sample, ${{Q}_{S}}$.

The system of equations (4), (5) and (6) can be solved to obtain expressions for the conductance of the legs, ${{G}_{L}}$ (assumed identical for both heaters) and of the NW sample, ${{G}_{S}}$:

$$\begin{eqnarray}&&{{G}_{L}}=\frac{{{Q}_{JH1}}+\frac{1}{2}{{Q}_{JL1}}}{\left( \Delta {{T}_{H1}}+\Delta {{T}_{H2}} \right)}+\frac{{{Q}_{JH2}}+\frac{1}{2}{{Q}_{JL2}}}{\left( \Delta {{T}_{H1}}+\Delta {{T}_{H2}} \right)}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{{G}_{S}}={{G}_{L}}\frac{\Delta {{T}_{H2}}}{\Delta {{T}_{H1}}-\Delta {{T}_{H2}}}-\frac{{{Q}_{JH2}}+\frac{1}{2}{{Q}_{JHL2}}}{2\left( \Delta {{T}_{H1}}-\Delta {{T}_{H2}} \right)}.\end{eqnarray} \tag{ 8 }$$

The difference to the previously reported [11] equations for ${{G}_{L}}$ and ${{G}_{S}}$ is that here the heating through the probing current ${{I}_{2}}$ is not neglected, giving rise to the second terms in (7) and (8). The better the devices are thermally decoupled, the more relevant the dissipation in the sensing heater becomes.

The Joule heat dissipated into the heaters is taken from the two-probe (${{R}_{2P}}$) and four-probe (${{R}_{4P}}$) resistance measurements, and we define an effective heater power (omitting the index for heater 1 and 2) as

$$\begin{eqnarray}&&{{P}_{{\rm eff}}}={{Q}_{JH}}+\frac{1}{2}{{Q}_{JL}}={{R}_{4P}}\;{{I}^{2}}+\left( {{R}_{2P}}-{{R}_{4P}} \right){{I}^{2}}.\end{eqnarray} \tag{ 9 }$$

The heat conductance measurements were performed with a dc technique. The heater (H1) was driven with constant heating current ${{I}_{H}}$ (swept typically between 0.1 μA and 10 μA) using the outer electrodes of the four-probe temperature sensor. The voltage drops across the inner (four-probe) and outer electrodes (two-probe) were measured simultaneously using two voltage meters (Keithley DMM 196). Two measurements with reversed polarity of ${{I}_{H}}$ were performed to eliminate thermoelectric offsets. In addition, ${{I}_{H}}$ was measured with a high-resolution electrometer (Keithley 6517) to improve accuracy. While applying a constant heating current to H1, the resistance of the four-probe temperature sensor (H2) was measured using a Keithley Delta Mode system consisting of a 6221 dc and ac current source and a 2182A nanovoltmeter. A constant sensing current ${{I}_{S}}$ of typically 100 nA was applied to the heater with a resistance in the 1–$10\;{\rm k}\Omega$ range, low enough to avoid self-heating of the sensor. The polarity of the current was reversed after a period of 200 to 500 ms to cancel out thermo-electric offsets. Afterwards, the Seebeck voltage across the NW was measured at each heater power using the nanovoltmeter connected to the inner NW contacts. Thermoelectric offsets of the circuitry were determined and subtracted as well. Subsequently, the roles of H1 and H2 were swapped and the measurement was repeated.

All measurements were performed in a cryo-probestation at a vacuum of $\approx 1\times {{10}^{-4}}\;{\rm Pa}$ (${{10}^{-6}}\;{\rm mbar}$). The temperature of the sample was measured with a calibrated Si-diode mounted on the surface of the test chip. Temperature stability of the sample was typically better than 0.1 K at any set temperature between 150 K and 350 K. Shielded electrical contacts to the test chip were provided by a 16-pin probecard connected to a 8 × 24 switching matrix (Keithley 7172).

The measurements of the thermal conductivity and the Seebeck coefficient require the precise determination of the temperature of the two heater platforms. Heater/sensor temperatures were determined from the TCR of the resistive coils measured by four-probe technique at a series of substrate temperatures. The current applied to read out the temperature, however, leads to an increase of temperature because of the high thermal resistance of the heater structures. On the other hand, the use of extremely small sensing currents induces only a marginal voltage drop across the heater and, hence, results in an inferior signal-to-noise ratio. Therefore, for calibration, the electrical resistances were determined in the limit of zero dissipated electrical power for each substrate temperature. It turned out, to a good approximation, that the electrical resistance increases linearly with the effective heater power. Therefore, the electrical resistance measured by extrapolating the linear resistance-power plot to zero corresponds to the temperature of the substrate. The standard deviation of typical measurement series was usually below 0.05 K.

Figure 4(a) shows the temperature increase in the heater and the sensor as a function of the effective heater power given in (9) at a substrate temperature of 300 K. Both directions (H1 as a heater, H2 as a sensor, and vice versa) are equivalent within an uncertainty of 0.1 K in the measured temperature difference.

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Having determined the temperature rise of heaters according to the calibration routine, the thermal conductances of the heater/sensor leads, ${{G}_{L}}$, and the NW sample, ${{G}_{S}}$, can be calculated using (7), (8) and (9). The data for each substrate temperature were averaged and are reported in figure 5. The thermal conductance of the measurement device (${{G}_{L}}$) is significantly lower than the resulting values for previously measured NWs [16]. Such devices are therefore promising for ultra-low thermal conductance measurements. The noise limit of the device and the measurement apparatus was estimated using the scatter of conductance data of the NW for a given substrate temperature because all measurement errors enter these data and are expected to induce scatter of the data. The one-sigma scatter of the data was below 0.2 nW ${{K}^{-1}}$.

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The thermal conductance of the heater/sensor legs, ${{G}_{L}}$, compares well with the expectations. For the particular design we used, the thermal resistance can be estimated from the dimensions. Two 300 μm long, 3 μm wide and 150 nm thick SiN membranes carry three metal lines with a width of 250 nm and a thickness of 50 nm. With typical values of thermal conductances of SiN (4 W ${{{\rm (m \cdot K)}}^{-1}}$) and Au (100 W ${{{\rm (m \cdot K)}}^{-1}}$) for thin films at room temperature [37, 38], we obtain 35 nW ${{K}^{-1}}$, in agreement with our experimental data. Other devices with leg lengths of up to 0.75 mm allowed the value of ${{G}_{L}}$ to be reduced down to 20 nW ${{K}^{-1}}$ at room temperature. The slight increase of the thermal conductance ${{G}_{L}}$ with temperature is expected for thin metal films. The NW used in this experiment had a length of 2.75 μm and a diameter of 125 nm. Its thermal conductance ${{G}_{S}}$ exhibited a weak  temperature dependence with a magnitude of about 12 nW ${{K}^{-1}}$ averaged over both signs of the temperature gradient.

The Seebeck coefficient (thermopower) was determined from the slope of the Seebeck voltage plotted against the temperature difference between the heater/sensor platforms as shown in figure 4(b). Here, both signs of the temperature gradient are plotted. The four-probe electrical resistance of the NWs was determined from measured current–voltage diagrams, taking the resistance value extrapolated to zero current. The extrapolation was done to ensure that the temperature of the wire is the temperature of the chip.