Letter pubs.acs.org/NanoLett
Large Thermoelectric Power Factor Enhancement Observed in InAs Nanowires Phillip M. Wu,*,† Johannes Gooth,†,‡ Xanthippi Zianni,§,∥ Sofia Fahlvik Svensson,† Jan Göran Gluschke,† Kimberly A. Dick,†,⊥ Claes Thelander,† Kornelius Nielsch,‡ and Heiner Linke*,† †
Solid State Physics and Nanometer Structure Consortium (nmC@LU), Lund University, Box 118, S-22100 Lund, Sweden Institute of Applied Physics, University of Hamburg, Jungiusstrasse 11, 20355 Hamburg, Germany § Department of Applied Sciences, Technological Educational Institution of Chalkida, 34400 Psachna, Greece ∥ IAMPPNM, Dept. of Microelectronics, NCSR ‘Demokritos’, 153 10 Athens, Greece ⊥ Polymer and Materials Chemistry, Lund University, Box 124, S-22100 Lund, Sweden ‡
S Supporting Information *
ABSTRACT: We report the observation of a thermoelectric power factor in InAs nanowires that exceeds that predicted by a single-band bulk model by up to an order of magnitude at temperatures below about 20 K. We attribute this enhancement effect not to the long-predicted 1D subband effects but to quantum-dot-like states that form in electrostatically nonuniform nanowires as a result of interference between propagating states and 0D resonances.
KEYWORDS: Nanowires, thermoelectrics, InAs, power factor, quantum dots, interference fficient thermoelectric energy conversion requires materials with low thermal conductivity κ and with a high power factor. Semiconductor nanowires have attracted great interest for thermoelectrics because phonon surface scattering substantially reduces their κ1−4 and because one-dimensional (1D) electron confinement effects are predicted to significantly enhance the power factor.5−9 However, this 1D power-factor enhancement effect has to date not been realized, to the very best of our knowledge, in part because very thin nanowires (typically less than 20 nm),5−7,10 careful control of the carrier concentration,11,12 and either high electron mobility or very short wires are required. Here, we show that these strict conditions do not pose a limit to the performance of nanowirebased thermoelectrics. We report the observation of a power factor in InAs nanowires that exceeds that predicted by a singleband bulk model by up to an order of magnitude at temperatures below about 20 K. We attribute this enhancement effect not to the long-predicted 1D subband effects but to quantum-dot-like states that form in electrostatically nonuniform nanowires as a result of interference between propagating states and 0D energy resonances.13,14 Our result represents the first observation of power-factor enhancement in nanowires due to quantum confinement effects. As this novel mechanism for power-factor enhancement is observed in field effect gated nanowires that are relatively thick (50−70 nm) and long
E
© 2013 American Chemical Society
(micrometer scale), and is tolerant to variations in intrinsic carrier concentration, it greatly increases the range of nanowire systems that may be used in thermoelectric applications. The long-standing prediction of an enhancement of the power factor, S2σwhere S is the thermopower and σ is the electrical conductivityin nanowires is based on the expected increase in the thermopower (defined as S = −Vth/ΔT, where Vth is the open-circuit thermovoltage in response to a temperature differential ΔT) due to the discontinuity of the 1D density of states.5 Together with the well-established reduction of κ in nanowires,2,3 an enhancement of S2σ would greatly increase the potential to achieve a breakthrough increase of the thermoelectric figure of merit Z = S2σ/κ.6 Power-factor enhancement in nanowires has previously been observed due to phonon-drag effects1,2 and due to increased mobility in core− shell wires.15 However, an enhancement of S2σ due to quantum-confinement effects in nanowires has to date not been reported. Such an enhancement due to 1D confinement effects is predicted for extremely thin nanowires, typically less than 20 nm,5−10 and requires careful control of the Fermi energy EF, which needs to be positioned accurately (within a Received: April 25, 2013 Revised: July 30, 2013 Published: August 6, 2013 4080
dx.doi.org/10.1021/nl401501j | Nano Lett. 2013, 13, 4080−4086
Nano Letters
Letter
thermal energy kT or better)11 just below the lowest occupied, 1D subband.5−7,11 Thermoelectric studies of 1D systems with control of EF have been carried out, for example in carbon nanotubes,16,17 as well as in systems of nanowires with different materials and different diameter, e.g., single-wire 100 nm PbSe,18 22 nm Sb3Te3,19 20 nm InAs,12 and 15−28 nm Si/Ge core−shell nanowires.15 There are several probable reasons for why a power-factor enhancement due to 1D confinement has so far not been observed:4 that the carrier mobility in nanowires is typically suppressed compared to bulk values;12,18 that the nanowires used were not sufficiently thin;10 that EF was not tuned to below the first subband;12,18−20 that the 1D subband structure was not resolved due to defects4,21 or due to a limited temperature range.15,17 In one of these works, Tian et al. attributed observed modulations of the power factor (similar to the ones reported here) as a function of EF in InAs nanowires to 1D effects, but an enhancement of S2σ was not observed. The question thus is whether a power-factor enhancement by quantum confinement can be achieved at all in nanowires and whether these nanoscale systems indeed can fulfill their potential in thermoelectric applications. Here we report power factors in nanowires that exceed by up to an order of magnitude those expected from single-band bulk models, under conditions where enhancement due to 1D effects is not expected. In particular, we observe high power factors in diffusive InAs nanowires with relatively thick diameters (D = 50−70 nm) and over a wide range of EF, which we tune using a global backgate. We attribute the powerfactor enhancement (observed below T ≈ 20 K) not to the long-predicted 1D confinement mechanisms but to quantumdot-like states that can form due to nonuniformities in the nanowires13,14 and which play a stronger role for electron transport at low values of EF. This represents the first observation of power-factor enhancement in nanowires due to quantum confinement effects. Our experimental analysis is in agreement with theoretical simulations that reproduce the dependence of the conductance G and Vth on EFwhich was tuned experimentally using the gate voltage, VG22at low temperatures (1.6−192 K) where confinement effects play a significant role in transport. The origin of the power-factor enhancement in quantum-dot-like states is supported by the fact that enhancement is observed for EF well above the first subband (instead of below the first subband as predicted by 1D models), by its nonmonotonic temperature dependence (which our simulations predict for nonuniform wires but is in contrast to 1D behavior), and by clear signatures of quantum-dot-like states in the nanowires’ G and Vth signals. Our experiments were performed using single-nanowire devices17 equipped with a global backgate for control of the nanowire EF; a resistive heater allowing a temperature differential ΔT to be generated over the length of the nanowire; two resistive thermometers, one at each end of the nanowire, for measurement of ΔT; and electrodes for the measurement of Vth and G (Figure 1a and Methods section). We present data for four InAs wires (denoted NW1−4) with diameters ranging from D = 50 to 69 nm and lengths L = 0.77 to 5.1 μm (see Methods for details). We chose this diameter range as the electrical properties of similar diameter nanowires have been studied previously in the literature3,23 and because their conductance is tunable (using a back gate) over a wide range. The use of several wires of similar width allows us to study the effect of sample-to-sample variations. Transmission
Figure 1. Device for thermoelectric measurement of single InAs nanowires. (a) Scanning electron microscope (SEM) image of a device for single-nanowire (highlighted in red) measurements. The n-doped Si/SiO2 chip served as a backgate. Bright lines are lithographically defined metal lines used as heater, as ohmic inner electrodes for electric measurements (see Figure S6), and as thermometers, as indicated. (b) Transmission electron microscope (TEM) image of a nanowire from the same growth substrate as those used in the measurements. The nanowires are wurtzite crystal phase with limited number of stacking faults (stacking faults indicated with white arrows). The scale bar is 50 nm.
electron microscope imaging (Figure 1b and Supporting Information Figure S7) of such nanowires (see Methods for growth details) reveals a wurtzite crystal structure with stacking faults on the order of 10 per micrometer. Our transport analysis showed that: the thermoelectric transport was mediated by electrons (all wires were depleted by negative VG, and we measured negative S); at the highest gate voltages used (VG ≈ 10−12 V) the nanowires had carrier mobilities μ on the order of 103 cm2/(V s) and carrier concentrations n in the range of 1017 cm−3, corresponding to a Fermi wavelength λF = 2π(6π2n)−1/3 ∼ 35 nm; at these large VG, where the wires are not depleted, we find S ≈ −10 μV/K, a value that is consistent with earlier measurements on InAs bulk material and nanowires (see Supporting Information and refs 3, 12, 24, and 25). We measure S and G as a function of VG for different substrate temperature T and calculate the gate voltage dependence of S2σ at these substrate T’s from the S and G data sets. Below VG ≈ 5 V we observe that |S| increases up to several mV/K as a function of decreasing VG (Figure 2a−c). This increase is expected when EF drops below the lowest subband11,13 and is here observed for the first time in nanowires. However, the large values of |S| are not accompanied by the (hitherto not observed) increase of the power factor S2σ predicted for the same gate-voltage range by 1D transport models for ideal nanowires:5−7,11 in our wires the product S2σ is negligibly small below the point where |S| increases (at about VG ≈ 5 V in Figure 2d). This is because G (used to calculate σ, see Methods) drops to very small values for decreasing VG (and decreasing carrier concentration) already before S diverges, preventing S2σ from rising. Large values of the power factor of S2σ are instead observed at higher gate voltages, where G is finite (see around VG ≈ 6 V in Figure 2d). The observed peak values greatly exceed, by up to an order of magnitude, those expected from a single-band model (see Methods) for a bulk material of similar quality at temperatures below about 20 K. This is shown in Figure 3a, where we plot peak power factor values for NWs 1−3 for each substrate temperature. Because the peaks in S2σ are observed well above the onset of the first subband, we do not attribute 4081
dx.doi.org/10.1021/nl401501j | Nano Lett. 2013, 13, 4080−4086
Nano Letters
Letter
Figure 2. Experimental and theoretical thermopower, conductance and power factor of InAs nanowires. Experimental (a−c) and theoretical (e−g) data showing thermopower (left axis) and conductance (right axis) as a function of VG at substrate temperatures as indicated for the nanowire sample NW1 (D = 50 nm) and the theoretical nanowire model shown in the inset to (g) (w = 50 nm, x = 25 nm, y = 100 nm, z = 25 nm, identical schematic shown in Figure S2). Similar behavior was seen experimentally for all wires studied. At low temperatures, modulations arising from quantum confinement effects are observed in S and G (a−c and e−g) that translate into peaks in the power factor (d, experimental; h, theoretical). Note that these fluctuations are not noise but are, in the experiments, fully reproducible within the same cooldown (Figure S1). Note also that the specific parameters chosen for the model are arbitrary and are intended to model a generic type of nonuniformity in a wire, regardless of the precise origin or geometry. The goal is to understand the qualitative implications of interference effects due to inhomogeneities/imperfections on the thermoelectric transport coefficients of nanowires. Additional scattering on impurities would enhance interference fluctuations in the conductance and further decrease its magnitude.31
the large S2σ values to the predicted, ideal 1D-subband effects.5−7,11 At the root of the S2σ peaks are fluctuations in G and S as a function of VG that are clearly visible in Figure 2a−c and that become more pronounced at low temperatures. Crucially, these fluctuations in G and S (and consequently in S2σ) are not noise: during the same cooldown, repeated gate voltage scans precisely reproduce the fluctuations in each parameter (Figure S1; note that fluctuations are not reproduced after thermal cycling to room temperature). Such fluctuations in G as a function of VG are commonly seen in nanowires,22 are thought to be due to a combination of scattering at wire imperfections and low-dimensional confinement, and can be interpreted as manifestations of features in the wire’s energy-dependent transmission function t(E). Sharp features in t(E) are known to enhance S because they can preferentially transmit charge carriers with high kinetic energy (energy filtering).5,7,26 As shown in more detail in Figure 4a−c, the fluctuations in S correlate to the derivative of G as expected from the Mott relation,27 and the observed peaks in S2σ can be clearly related to the combined effect of fluctuations in S and G (Figure 4d).
More specifically, we propose that quantum-dot-like states (whose effect becomes more pronounced near depletion, at low VG) in the nanowires cause the observed fluctuations in G, S, and S2σ,13,14 based on two experimental observations: first, our measured S shows unique sign reversals as a function of VG that are apparent both in Figure 2a (at VG ≈ 6 V) and Figure 4c (at VG ≈ 3.6 V) and that are characteristic to quantum dots with sharp resonances of t(E).28,29 Second, a 2D map of the wire’s differential conductance as a function of Vsd and VG (a stability diagram, see Figure 4e) reveals Coulomb diamonds 22 (signatures of quantum dots resonances) that clearly correlate with the peaks in S2σ (Figure 4d,e). Note that in a uniformly 1D nanowire such Coulomb diamonds are expected to be absent. Very well-defined diamonds, on the other hand, would be expected only for quantum dots that are essentially decoupled from the 1D leads (for example, by heterostructure barriers).30 Weakly defined diamonds such as the ones visible in Figure 4e occur when the quantum-dot-like states are strongly coupled to the electron reservoirs (or contacts). Such states can form in nanowires due to the interference between propagating states and energy resonances that form in nonuniformities of the wire.13,14 They lead to sharp energy structure of t(E) that 4082
dx.doi.org/10.1021/nl401501j | Nano Lett. 2013, 13, 4080−4086
Nano Letters
Letter
Figure 3. Enhanced thermoelectric power factor in InAs nanowires as a function of temperature (logarithmic scale). (a) Measured peak power factor obtained by taking for each substrate temperature the peak value of the power factor as a function of gate voltage (resolution 1 mV). Note that peak power factor occurs at different gate voltages (see Figure 2d) for each temperature and for each wire, and the absolute magnitude will vary from wire to wire. By showing data from several wires we offer a feel for sample-to-sample variation. The dashed lines show the expected power factor based on a simple parabolic-band bulk model using a carrier-concentration independent mean free path (see legend and Methods).36 Whereas at higher temperatures the measured S2σ roughly follow this bulk model and decrease with decreasing temperature, the experiments show a significant increase below about 50 K that peaks at about 6−10 K. (b) Calculated temperature dependence of the peak power factor for the model wire with modulated width as shown in inset of Figure 2g, at different EF as indicated. Note that also in the model power-factor peak values and peak temperatures depend on model details, such as the EF chosen. However, importantly, in both experimental and theoretical data, the temperature dependence is nonmonotonic, in contrast to what is expected for a 1D, and both predict and show power-factor enhancement in the same temperature range.
nonuniformities are expected to reduce G.31 We therefore focus on a qualitative comparison of theory and experiment. Importantly, the calculations reproduce and explain several key features of the experimental data. First, at low energies (near depletion) quantum-dot-like states in the model lead to the same characteristic28 sign reversal of S (Figure 2e, EF ≈ 40 meV) that is also visible in the experiment (Figure 2a, VG ≈ 6 V). Second, quantum-dot-like states in the theory are responsible for the suppression of G with decreasing EF, making G vanishingly small already before EF falls below the lowest subband (EF = 0) and where S diverges. This prevents the dominant peak in power factor near depletion that is characteristic to the ideal 1D models5,11 (Figure S3) but that is not visible in theory or experiment in Figure 2. Third, the interference-related fluctuations in the calculated G, S, and S2σ are smeared out at higher temperatures (Figure 2c,g). This suppression of the S2σ peaks at higher temperatures leads to a nonmonotonic temperature dependence of the power factor observed in experiment and theory (Figure 3), which is in contrast to the monotonic T-dependence of S2σ for ideal 1D transport (Figure 3b), and can thus be viewed as a fingerprint of quantum-dot-like states. Our results represent the first experimental demonstration that quantum-confinement effects in nanowires can enhance the thermoelectric properties. The same generic, qualitative behavior can be identified in both the experiment and in calculations of nonuniform wires. In both, we observe features
are in addition to those of the 1D-subband structure (Figure S2) and have been predicted to enhance the thermoelectric performance.13,14,26 Our observation of a diamond-like stability diagram (Figure 4e) that correlates with the thermoelectric data strongly supports the proposed role of quantum-dot-like states. To compare the effect of such quantum-dot-like states to our observations, we performed calculations of the transport properties of nanowires with inhomogeneities (Figure 2e−h) as well as of uniform nanowires (Figure S3). There are a number of possible types of inhomogeneities that may, in the experiment, give rise to quantum-dot-like states, including impurities, a nonuniform diameter, surface charges, or stacking faults. However, from the quantum-mechanical point of view, each has the same effect: to modify the carrier’s energy spectrum through quantum interference. To qualitatively compare the expected role of such interference effects for the thermopower, we here choose to effectively model such a inhomogeneity as a nonuniform nanowire, using the specific (but arbitrary) geometry shown in Figure 2g. Although our nanowires are in the diffusive transport regime, the calculations were performed in the ballistic transport regime and provide qualitative insights into the experimental results. This approximation has the important advantages that it is transparent to interpretation and that the thermoelectric transport coefficients can be directly related to t(E) but, as a side effect, leads to a theoretical G that is much higher than in the experiment (Figure 2), where scatterers additional to 4083
dx.doi.org/10.1021/nl401501j | Nano Lett. 2013, 13, 4080−4086
Nano Letters
Letter
Figure 4. Thermopower and Coulomb blockade in InAs nanowires. (a) The conductance G of NW4 (D = 69 nm) as a function of back-gate voltage, VG. The conductance trace shows two particularly distinct peaks at approximately 2.5 and 3.3 V that we interpret as corresponding to quantum-dotlike states. (b) The derivative of the conductance trace shown in (a). (c) The thermovoltage Vth of the same nanowire as a function of VG at constant heating power. The thermovoltage reproduces the features of the derivative of the conductance, shown in (b), as described by Mott’s formula.27 Note the sign reversal in thermovoltage at a back-gate voltage of approximately 3.6 V, a feature characteristic to 0D states. The traces shown in (a) and (c) were measured in separate sweeps, with negligible hysteresis effects (Figure S1). (d) The conductance, shown in (a), multiplied with the thermovoltage, shown in (c), squared (proportional to the power factor) shows features that clearly line up with the quantum-dot-like states. (e) To further illustrate the quantum-dot-like nature of the power-factor peaks, source-drain sweeps at constant back-gate voltages ranging from 2 to 4.5 V (stability diagrams) have been conducted. The emerging structure (white dotted lines) strongly resembles Coulomb blockade diamonds: a signature of quantum dots. All data shown were measured at a base temperature of 4 K.
that are typical to quantum dots, namely fluctuations and Coulomb diamonds in G and characteristic sign changes in Vth, as well as a nonmonotonic temperature dependence of the power factor. We therefore attribute the observed enhancement to quantum-dot-like states in the nanowires. We now address the expected generality of our results. First, power-factor fluctuations of the type observed and reported here are expected in any nanowire that can be fully depleted (using a gate) at sufficiently low temperature, where interference effects are pronounced. This is expected because nanowires very generally show fluctuations in G(VG), of the type that give rise to fluctuations in S and thus in S2σ, and that are attributed to wire nonuniformities. From a modeling perspective, we then also expect the same qualitative behavior regardless of the precise material, shape, or length of the wire. However, the magnitude of the power-factor fluctuations, as well as the values of VG or T where peak values are observed, will depend on fine details of the nanowires (including carrier mobility, material type, impurities, surface charges, and diameter modulations), which typically vary from sample to sample, and whose role should be explored in future experiments. Future theory should take into account the effects of diffusive transport.
Important for future applications, for example in solid-state cooling, we observed the enhancement in wires that are much thicker than the very thin (
