Very High Thermopower of Bi Nanowires with Embedded Quantum

Letter pubs.acs.org/NanoLett

Very High Thermopower of Bi Nanowires with Embedded Quantum Point Contacts Eyal Shapira,† Amir Holtzman,† Debora Marchak, and Yoram Selzer* School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel ABSTRACT: Quantum confinement effects in bismuth (Bi) nanowires (NWs) are predicted to impart them with high thermopower values and hence make them efficient thermoelectric materials. Yet, boundary scattering of charge carriers in these NWs operating in the diffusion transport regime mask any quantum effects and impede their use for nanoscale thermoelectric applications. Here we demonstrate quantum confinement effects in Bi NWs by forming in their structure ballistic quantum point contacts (QPCs) leading to exceptionally high thermopower values (S > 2 mV/K). The power factor, S2G, of the QPCs is maximized at G ∼ 0.25G0 (where G0 is the quantum of conductance) within agreement with a one-band model with step edge characteristics. KEYWORDS: Thermopower, Seebeck coefficient, quantum point contact, quantum confinement, nanowires

T

Here we demonstrate quantum confinement effects by exploiting the long electron mean free path (100 nm at 300 K and ∼0.4 mm at 4 K) of Bi and by fabricating NWs with embedded ballistic quantum point contacts (QPCs). QPCs are short conducting constrictions with a characteristic width comparable to the Fermi wavelength of their charge carriers, characterized by quantized conductance at integer multiples of G0 = 2e2/ℏ, where e is electron charge and ℏ is the Plank constant.9,10 On the basis of the Landauer-Büttiker formalism, the thermopower in the ballistic regime is approximately (see further discussion below)

hermoelectric (TE) materials convert thermal gradients and electric fields for power conversion and for refrigeration, respectively.1,2 TE devices are all-solid-state units with no moving fluids and mechanical parts and thus are highly reliable and ideal for integration into small scale applications with moderate power demands. Their performance is limited to a fraction of the Carnot efficiency and is described by a figure of merit ZT defined as

ZT =

S 2GT κ e + κL

(1)

where S is the Seebeck coefficient (thermopower), T is an average temperature, G is the electrical conductance, and κel, κL, are the electronic and lattice components of the thermal conductance, respectively. Currently state of the art devices have ZT ∼ 1.5, making them even less efficient than household refrigerators. It has been proposed that band structure engineering by the introduction of low dimensional (2D and 1D) structures, such as superlattices and nanowires (NWs) can enhance the performance of thermoelectric materials.3,4 Specifically, high ZT values have been theoretically envisioned for bismuth (Bi) NWs.5,6 Bi has a long electron Fermi wavelength (λF ∼ 26 nm) and therefore it should be easy to make Bi NWs that reach the 1D conduction limit6 with this material. It also has low bulk thermal conductivity (8 W m−1 K−1), which is expected to become even lower in a NW geometry due to acoustic phonons confinement and scattering with boundaries.7 While high thermopower values have been measured for two-dimensional electron gas in SrTiO3,8 realization of similar values due to quantum confinement effects in individual and well characterized NWs have never been achieved. © 2012 American Chemical Society

E1 − EF (2) eT where E1 is the threshold energy of the lowest one-dimensional sub-band or mode in the QPC, and EF is the Fermi energy. QPCs based on GaAs two-dimensional electron gas structures have a typical band gap in the order of tens of microvolts, and therefore could have high S values only at very low temperatures (T < 1 K).10 Short molecular junctions operating in the ballistic regime with a HOMO−LUMO gap in the order of 1 eV, could potentially have a high S value in the order of mV/K at room temperature.11 However, current lack of capability to effectively tune the Fermi energy position within their gap prohibits maximization of S2G and hence their use for thermoelectric applications. Ballistic Bi constrictions with bandgap in the hundreds-of-millivolts range offer the possibility S≈−

Received: November 1, 2011 Revised: December 12, 2011 Published: January 3, 2012 808

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Figure 1. (a) A low-magnification TEM image of a Bi nanowire. Inset: A high-resolution TEM image of the nanowire showing a single-crystal structure. The FFT pattern along the [001] zone axis indicates that the growth direction of the nanowire is [110]. (b) Resistance of a typical 40 nm NW as a function of temperature measured by 2 and 4 probes. Assuming R ∼ R0 exp(ΔEg/2kBT), where ΔEg is the band gap, we find for all NWs, ΔEg = 12 ± 3 meV, which is within agreement with the theoretical band structure for NWs with this diameter.

Figure 2. (a) A device for conductance and thermopower measurements (see text for details). Inset: A QPC formed by electromigration. (b) Change in conductance of a NW as a QPC is formed by electromigration at 77 K using a feedback controlled scheme (see ref 22). The resulting QPC at the end of the curve with conductance in the tunneling regime (G < G0) is shown in the inset image in (a). Inset: Theoretical band structure of a NW as a function of diameter (see ref 6). The dashed line is the position of the Fermi energy in the middle of the gap of a 40 nm NW, assuming intrinsic doping conditions.

ultimately to fully oriented wires in the [110] direction for diameters ≤40 nm. Figure 1b demonstrates that ohmic contacts to selected NWs can be established once the oxide layer on the wires is removed. The plot shows identical resistance values for a certain NW measured by both a 2- and 4- probes method over the temperature range 77−300 K. Devices for conductance and thermopower measurements are patterned on selected NWs as depicted in Figure 2a.15 The two resistors, Rn and Rf serve a dual purpose to probe the thermovoltage across the NW under a temperature gradient and to serve as thermometers that determine the temperature gradient across the NW by accurately measuring changes in their resistance. A microheater, Rh, is fabricated adjacent to one end of the NW. In a typical experiment, a heating current Ih is applied to the heater forming a temperature gradient (typically ∼0.5 K) across a segment of the NW. The resulting thermovoltage Vp is determined by measuring the potential across the segment, enclosed by Rn and Rf, by a high impedance voltmeter. The thermopower is calculated from S = Vp/ΔT. The presented values of S are corrected by subtracting the thermovoltage on the Au thermometers/probes, assuming they

to achieve high thermopower at temperatures between the above two extremes. Fabrication of ballistic devices starts by synthesizing Bi NWs with a typical diameter of ∼40 nm using a previously described procedure.12 Briefly, a U-shaped glass tube comprising of a 6 μm thick porous polycarbonate (PC) membrane with a monodispersed distribution of pores with an average diameter of 10 nm is positioned between two aqueous solutions, one containing Bi2Cl3 and the second containing the reducing agent NaBH4. The two solutions intermix within the pores of the membrane and react to form Bi NWs. The crystal structure of the NWs was determined by transmission electron microscopy (TEM) as shown in Figure 1a. The HRTEM and the corresponding FFT image show the atomic fringes of the wire demonstrating a single-crystalline character. The growth front is assigned to be [110]. Our results are consistent with previous studies13,14 that show that electrodeposition of Bi into pores under potentiostatic conditions (essentially similar to our process) results in wires oriented preferentially along the [110] direction. This tendency increases as the diameter of the wires decreases, leading 809

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possess bulk Au thermopower properties. The temperature gradient is determined by using a standard 4-probe lock-in technique to measure changes in the resistances of Rn and Rf. These changes are related to the increase in local temperature by the general formula R = R0(1 + βΔT) where R0 is the resistance at a given reference temperature, taken here as 300 K, ΔT is the temperature change from the reference temperature, and β (=dR/dT) is the temperature coefficient of resistance (TCR). The value of β for each thermometerresistor was determined prior to the experiments by varying the temperature of the cryostat, and was found to be 1.4 × 10−3 K−1, which is in agreement with previous results for thin films resistors.15 All measurements were performed under vacuum (10−6 Torr) and at 77 K where the electronic properties of Bi are well-defined,6 allowing as described below quantitative analysis and modeling of the results. The thermopower of the NWs, prior to QPCs formation, is −20 ± 5 μV/K suggesting, according to the sign, electrical transport properties dominated by electrons, which is in par with previous studies.16 Electromigration (EM) is employed to form QPCs in the NWs by introducing constrictions into their structure.17 EM is the directed migration of atoms caused by momentum transfer from electrons to atoms under large electric current density. As a constriction is formed and narrowed the decreasing conductance, G, is monitored and the controlled procedure allows termination of the EM process once a chosen conductance is achieved. A typical trace of conductance as a function of time is depicted in Figure 2b. Subsequent imaging by a scanning electron microscope (inset Figure 2a) reveals that although the final conductance is in the tunneling regime (G < G0) the formed constriction is still in mechanical contact without any apparent gap. This can be understood by considering the theoretical band structure as a function of diameter in Figure 2b. As the QPC diameter d decreases the quantized transversal momentum of the charge carriers kn ∝ 1/d increases.18 This leads to a shift in the subband energy position, En ∼ ℏ2kn2/2m* (larger for light carriers) and the gap between light and heavy electrons increases. An effective barrier is thus formed in mechanical contact and is related to the electron confinement in the constriction. In a free electron picture, the longitudinal momentum of each transversal mode or channel n is given by kzn = (KF2 − kn2)1/2 (with KF = 2π/λF) such that to each channel n corresponds a 1D effective potential barrier in the longitudinal direction. The modes with kn > KF are evanescent or tunneling modes. When d < λF(26 nm) there is no room even for a single channel and the conductance is dominated by electron tunneling through the first channel. Figure 3 depicts two representative EM-induced conductance traces where at certain conductance values the thermopower of the formed QPCs was measured as well. When G > G0, the thermopower is negative and has a typical value of −20 μV/K. Once G < G0, as a general trend the thermopower changes sign (S > 0) and also increases by more than an order of magnitude. The apparent noise in the traces in Figure 3, which in the right panel appears spikes of conductance, is due to the method by which the QPCs are formed. We attribute these spikes to abrupt structural changes in the contacts, as part of the structural annealing that is taking place following the voltage pulses that initiate electromigration.17

Figure 3. Thermopower and conductance traces. QPCs are formed in the NWs by applying voltages in a square wave pattern comprises of 200 ms of a high bias, typically 0.3 < V < 1.3 to induce EM, followed by 200 ms of 1 mV in order to determine the resulting conductance (see ref 17). The latter values are the plotted conductance values in all graphs. The process is repeated until a desired conductance value is approximately reached. At this point the EM-inducing circuit is disconnected and the Seebeck coefficient is determined after which the EM process is resumed.

The initial negative value of S results from the much higher mobility of electrons than holes. As the band structure in Figure 2 suggests, QPC formation is accompanied by a transition from transport dominated by light (L-pocket) electrons carriers to (T-pocket) hole carriers at small cross sections with less population of the light electron states. This change is accompanied by a change in the sign of S. The large change in the absolute magnitude of S is appreciated by considering the expressions for G, and S according to the Landauer-Büttiker formalism10,19,20

G=−

2e 2 h

∂f

∫ dE ∂E τ(E)

(3)

∂f

1 ∫ dE ∂E τ(E)(E − EF) S= ∂f eT ∫ dE τ(E) ∂E

(4)

where τ(E) is the energy dependent transmission function, and f is the Fermi distribution, f(ε) = [exp[(E − EF)/(kBT)] + 1]−1 . We approximate τ(E) to be a step function of the form ∞

τ(E) =

∑ θ(E − En) n=1

(5)

where the steps in τ(E) coincide with the threshold energies En of the one-dimensional sub-bands or modes in the QPC. Figure 4 plots theoretical curves of G and S as a function of (E − Emidgap)/kBT for different constriction diameters where Emidgap is the midgap energy. The position of the band edges at each diameter is according to the band structure in the inset of Figure 2b. The thermopower is zero when E = Emidgap, due to symmetric contribution of the conduction and valence subbands. It is also zero inside the bands where the derivative of f is nullified. In between, according to eq 3, a peak in S is at the energy where the transmission-weighted average value of the heat energy E − EF is maximized. With decreasing size of the constriction, the band gap and the absolute value of the peak in 810

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Figure 4. Theoretical calculation of S (blue) and G (black) for several constriction diameters revealing a qualitative picture of increasing S with decreasing diameter and large positive S at the Fermi energy (marked by red dashed line).

S increase. The position of the Fermi energy of the system is marked and is determined by the rest (and much larger part) of the system outside the constriction (at Emidgap for a 40 nm NW assuming intrinsic conditions). According to the model, the thermopower of the formed QPCs should be positive and is progressively becoming larger with decreasing constriction size and conductance. Substituting eq 4 into 2 and considering only one (hole) band edge at E1, the conductance and the thermopower at the Fermi level can be described by10

G(EF) =

2e 2 × h

1 ⎛E −E ⎞ 1 + exp⎜ 1 F ⎟ ⎝ kBT ⎠

k ⎛ E − EF ⎞ S ≈ − B ⎜1 + 1 ⎟ e ⎝ kBT ⎠

(6)

Figure 6. Plot of the power factor S2G as a function of G. The red curve is a lead to the eye. A maximum value is reached at G ∼ 0.25G0 within agreement with the theoretical ballistic model (see inset).

(7)

5 and 6), the power factor is maximized at G ∼ 0.25G0. Indeed we find in Figure 6 that our experimental results support this expected behavior quite well. The maximum power factor in Figure 6 can be converted to 0.001 W m−1 K−2, by assuming a mean free path of the charge carriers within the order of the constriction’s diameter (∼10 nm, see right panel in Figure 4). This power factor is comparable to the one reported for highly doped Si nanowires at 77 K23 but is smaller than recent values reported for Si nanoribons.24 To summarize, we have demonstrated quantum confinement effects on the thermopower of Bi QPCs. As a result thermopower values higher than 2 mV/K were measured at 77 K. The measured values can be quantitatively explained by a one-band approximation of the QPC with step function characteristics. Work is currently under progress to determine the temperature dependence of the thermopower as well as the thermal conductance of the QPCs in order to fully characterize their performance as thermoelectric elements.

Figure 5 shows S as a function of G based on many traces similar to those described in Figure 3. The theoretical fit in this

Figure 5. Plot of S as a function of G extracted from traces like in Figure 3. The red line is the expected theoretical ballistic behavior (see text for details).

figure is based on calculating E1 − EF for each G value using eq 5, and then plugging E1 − EF into eq 6 to calculate S. The apparent good fit supports the dominating one band assumption and the step function characteristics of the transmission. Figure 5 shows that exceptionally high thermopower values (S > 2 mV/K) can be achieved with Bi QPCs using quantum confinement effects. Comparable values at cryogenic temperatures were measured in doped Ge, although in this case the high values originate in the bands-mass21,22 and not from confinement effects. For best performance, it is important to optimize the power factor, S2G, of the QPCs (see eq 1). The inset in Figure 6 shows that theoretically according to the one band model (eqs

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions †

These authors contributed equally to this study.

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ACKNOWLEDGMENTS

This research has been supported by Clal Biotechnology Inc. and by the Gordon Center for Energy studies in Tel Aviv University. 811

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