LETTER pubs.acs.org/NanoLett
A Nanoscale Standard for the Seebeck Coefficient Preeti Mani,† Natthapon Nakpathomkun,‡ Eric A. Hoffmann,§ and Heiner Linke*,|| Department of Physics and Materials Science Institute, University of Oregon, Eugene, Oregon 97403, United States ABSTRACT: The Seebeck coefficient, a key parameter describing a material’s thermoelectric performance, is generally difficult to measure, and no intrinsic calibration standard exists. Quantum dots and single electron tunneling devices with sharp transmission resonances spaced by many kT have a material-independent Seebeck coefficient that depends only on the electronic charge and the average device temperature T. Here we propose the use of a quantum dot to create an intrinsic, nanoscale standard for the Seebeck coefficient and discuss its implementation. KEYWORDS: Thermoelectrics, Seebeck coefficient, thermopower, quantum dots, measurement standard
A
material’s thermoelectric response to an applied temperature differential, ΔT, is called its thermopower and is described by the Seebeck coefficient S = VOC/ΔT, where VOC is the open-circuit voltage across the material. Together with the material’s electrical and thermal conductivities, S determines the performance of thermoelectric materials. There is great current interest in the development of advanced, nanostructured, and low-dimensional thermoelectric materials,1 and accurate metrology of S is therefore of significant interest. Physical calibration standards for S based on well-characterized bulk materials are currently being developed.2 However, based on cross-tests performed in multiple laboratories, their accuracy is currently limited to about (10 μV/K, or to about 10% of typical values of S.2 It is therefore desirable to develop a material-independent (intrinsic) standard against which reference materials can be calibrated. Intrinsic standards based on nanoscale quantum effects used in other metrological applications achieve accuracies better than 1 ppm.3 Examples include the Josephson junction based standard for voltage,4 the quantum Hall effect as the standard for resistance and the fine-structure constant,5 and single-electron tunneling (SET) devices used as the standards for capacitance6 and current7 and for primary thermometry.8 Quantum dots with well-spaced and very sharp transmission resonances have a Seebeck coefficient that is material-independent9,10 and are thus candidates for intrinsic, nanoscale standards for the Seebeck coefficient.11 Here, we discuss the accuracy that can be achieved in devices with finite broadening of the transmission resonances, develop a method to correct for nonidealities observed related to such broadening, and discuss the implementation of a nanoscale quantum dot standard. We consider the generic system illustrated in Figure 1. A twolead quantum dot is connected by tunneling barriers to a cold and a hot electron reservoir with electrochemical potentials μc/h = μ ( eV/2 and temperatures Tc/h = T - ΔT/2, respectively. Here μ and T are the average electrochemical potential and temperature r 2011 American Chemical Society
Figure 1. Key parameters defining the energy levels in a double-barrier quantum dot embedded between a hot and a cold electron reservoir, and biased by a voltage V = (μc μh)/e. When ΔE . kT, only one energy level contributes to electron transport across the dot.
of the two reservoirs, respectively, and V and ΔT are the bias voltage and temperature differential, each assumed to be applied symmetrically across the dot. The dot’s transmission resonances are separated in energy by ΔE, given by the dot’s charging energy and (for very small dots) by quantum confinement effects. We consider the case where ΔE . kT and where μc and μh are approximately aligned with one specific transmission resonance at energy Eres, such that only this one resonance contributes to electron transport. The energy-dependent transmission function of such a system is described by the Lorentzian12 τðEÞ ¼ A
ðΓ=2Þ2 ðE Eres Þ2 þ ðΓ=2Þ2
ð1Þ
characterized by its full width at half-maximum (fwhm) Γ and amplitude A. Finally, we assume sufficiently low temperatures (usually less than about 10 K) such that electronelectron and electron phonon interactions are negligible and transport across the dot Received: July 4, 2011 Revised: October 4, 2011 Published: October 17, 2011 4679
dx.doi.org/10.1021/nl202258f | Nano Lett. 2011, 11, 4679–4681
Nano Letters
LETTER
is elastic. The two-terminal current is given by the Landauer equation13 Z 2e ½fc ðE, μc , Tc Þ fh ðE, μh , Th ÞτðEÞ dE I ¼ ð2Þ h where fc/h = [1 + exp βc/h]1 with βc/h(E) = [(E μc/h)/kTc/h] are the FermiDirac distributions of the cold and the hot reservoir, respectively. To calculate S one needs to determine VOC, the bias voltage at which I = 0. Initially, we consider the ideal case of an infinitely sharp transmission resonance (Γ f 0) centered at energy Eres such that τ(E) limits to a delta function, τ(E) µ δ(E Eres). In this limit, one finds that I = 0 when βc(Eres) = βh(Eres). Substituting in this equation VOC = (μc μh)/e and ΔT = Th Tc one obtains the Seebeck coefficient S = VOC/ΔT of an ideal quantum dot (Γ f 0)9 ðEres μÞ ð3Þ eT This equation is the key point of this paper: Sideal depends only on T = (Th + Tc)/2, and the energy difference (Eres μ), and is independent of material parameters. We now consider the effect of finite broadening Γ of a dot’s Lorentzian transmission resonance. In the limits eV , kT and ΔT , T,14 one can expand fc and fh about μ and T ∂f0 eV ΔT ðE μÞ ( fc=h ¼ f0 þ ð4Þ 2 2 T ∂E Sideal ¼
Figure 2. Sideal as given by eq 3 (solid, red line) and SΓ as given by eq 5 for Γ = 0.001kT (circle, green curve), Γ = 0.01kT (dots, blue curve), and Γ = 1kT (dashes, brown curve), graphed for T = 1.05 K and ΔT = 0.1 K. Typical shapes of FermiDirac distributions fc and fh (upper inset) as well as Δf (lower inset) are graphed here using the values T = 1.25 K, ΔT = 1.5 K, V ≈ 0.1 mV.
where f0 = [1 + exp(β)]1 with β = [(E μ)/kT]. By substituting eq 4 into eq 2 and solving for I = 0, one finds Z ∂f0 ðE μÞτðEÞ dE 1 ∂E Z ð5Þ SΓ ¼ ∂f0 eT τðEÞ dE ∂E Note that in the limit Γ f 0, where τ(E) µ δ(E Eres), eq 5 reduces to eq 3 predicting a linear dependence of SΓ on (Eres μ), shown as the solid line in Figure 2. For finite Γ, eq 5 needs numerical evaluation. In Figure 2, the curves a, b, and c show eq 5 for T = 1.05 K and Γ = 0.001kT, 0.01kT, and 1kT, respectively. Compared to Sideal, SΓ has a smaller slope as a function of (Eres μ) near (Eres μ) = 0, and falls off to zero for large (Eres μ). To understand these differences, note that the current through the dot is determined by Δf(E) = (fh fc) convoluted with τ(E) (see eq 1 and insets to Figure 2). Consider an experiment where ΔT is fixed, (Eres μ) is controlled using a gate voltage, and μc floats relative to μh under open-circuit conditions. For Γ = 0, the system will self-adjust for all (Eres μ) such that Eres = Es, where Es = (μcTh μhTc)/(Th Tc) is the energy for which Δf = 0 for finite ΔT. This is the behavior predicted by eq 3. For finite Γ, τ(E) is a Lorentzian with long tails that sample Δf away from Eres. In general, Δf is not antisymmetric about Es (see Figure 2, lower inset) and for the zero-current condition to be fulfilled, Eres must be positioned away from Es unless (Eres μ) = 0. Therefore, the slope of SΓ near (Eres μ) = 0 depends on Γ. For large (Eres μ) and finite Γ, the leakage current through the tail of τ(E) at high bias becomes large, such that no significant VOC can be maintained by a temperature differential, and SΓ goes to zero. To quantify the difference ΔS = (Sideal SΓ), we define in the inset to Figure 3 the offset F as the absolute minimum of
Figure 3. Inset: (ΔS/Sideal) as a function of (Eres μ) for T = 1.05 K and Γ = 80 μeV (Γ/kT ≈ 0.9). The offset F is defined as the absolute minimum of (ΔS/Sideal). Main figure: numerically calculated values of F as a function of Γ/kT for Γ ranging from 0.8 to 100 μeV and T ranging from 2.1 to 525 K depend only on Γ/kT and are well fit by eq 6.
ΔS/Sideal, which occurs in the limit (Eres μ) f 0. By graphing F over a wide range of values of Γ and T (Figure 3) we find numerically that F depends only on the ratio Γ/kT and can be approximated by F ≈ 0:044ðΓ=kTÞ2 þ 0:27ðΓ=kTÞ
ð6Þ
allowing one to estimate this systematic error and correct for it if necessary. In a calibration experiment, one would use a gate voltage Vg to control the chemical potential of the source and drain leads relative to a quantum dot’s resonant energy and measure the dot conductance in order to obtain information about Γ.10 The difference Eres μ is related to the gate voltage by Eres μ = αe(V0 Vg), where α is the lever arm of the device.15 The measured Seebeck coefficient is then compared to Sideal expressed in terms of gate voltage, Sideal = α (V0 Vg)/T, where V0 is gate voltage at which the measured Seebeck coefficient is zero. ΔS/Sideal can be graphed against α (V0 Vg), and F determined by evaluating the minimum of ΔS/Sideal, for example 4680
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Nano Letters using a parabolic fit in the vicinity of Vg = V0. For a weakly coupled quantum dot (thick tunneling barriers), one can expect Γ , kT for T ≈ 1 K and F , 1. A value of F that exceeds eq 6 would indicate a problem with the measurement setup, such as the thermometry, the measurement of the open-circuit voltage, or parasitic effects, allowing the use of a quantum dot for system calibration. A back-gated quantum dot system as discussed here can be realized, for example, using a double-barrier structure embedded into a semiconductor nanowire,16 a configuration for which thermometry is available17 and which may be useful for the calibration of microstructures that measure S of suspended nanowires.18 Indeed, recent experiments using heterostructure nanowires confirm that eq 5 accurately predicts a quantum-dot’s thermopower, using measurements of the dot’s conductance and temperature as the only required input.10 Any effects of the measurement setup (such as the use of a finite load-resistance) on the measured S can be very well accounted for.10 Alternatively, one can consider an on-chip implementation using either a quantum dot19 with integrated thermometry realized in a two-dimensional electron gas or a tunnel-junction single-electron transistor.20 The latter system may be integrated with Josephson junctions for primary voltage measurement4 and tunnel junctions for primary thermometry,8 such that a Seebeck nanoscale standard can be designed, in principle, to dovetail nicely with existing quantum metrology. Because of the relatively small level spacing in semiconductor quantum dots, the condition ΔE . kT used here is not likely to hold at temperatures much above 10 K. For use near room temperature, it is worth considering the use of molecular junctions, which can have level spacings much larger than those in quantum dots and for which thermopower theory for use at room temperature is now becoming available.21 With an expression for the thermopower as a function of chemical potential in hand, it should be possible to account for the expected deviation from Sideal using a similar numerical procedure as the one employed here (Figure 3).
’ AUTHOR INFORMATION
LETTER
(2) Lowhorn, N. D.; Wong-Ng, W.; Zhang, W.; Lu, Z. Q.; Otani, M.; Thomas, E.; Green, M.; Tran, T. N.; Dilley, N.; Ghamaty, S.; Elsner, N.; Hogan, T.; Downey, A. D.; Jie, Q.; Li, Q.; Obara, H.; Sharp, J.; Caylor, C.; Venkatasubramanian, R.; Willigan, R.; Yang, J.; Martin, J.; Nolas, G.; Edwards, B.; Tritt, T. Appl. Phys. A: Mater. Sci. Process. 2009, 94, 231. (3) Flowers, J. Science 2004, 306, 1324. (4) Benz, S. P.; Dresselhaus, P. D.; Burroughs, C. J. IEEE Trans. Instrum. Meas. 2001, 50, 1513. (5) von Klitzing, K.; Dorda, G.; Pepper, M. Phys. Rev. Lett. 1980, 45, 494. (6) Zimmerman, N. M.; Keller, M. W. Meas. Sci. Technol. 2003, 14, 1237. (7) Clark, A. F.; Zimmerman, N. M.; Williams, E. R.; Amar, A.; Song, D.; Wellstood, F. C.; Lobb, C. J.; Soulen, R. J. Appl. Phys. Lett. 1995, 66, 2588. (8) Pekola, J. P.; Hirvi, K. P.; Kauppinen, J. P.; Paalanen, M. A. Phys. Rev. Lett. 1994, 73, 2903. (9) Beenakker, C. W. J.; Staring, A. A. M. Phys. Rev. B 1992, 46, 9667. (10) Svensson, S. F.; Persson, A. I.; Hoffmann, E. A.; Nakpathomkun, N.; Nilsson, H. A.; Xu, H. Q.; Samuelson, L.; Linke, H. http://arxiv.org/ abs/1110.0352, 2011. (11) Mani, P.; Nakpathomkun, N.; Linke, H. J. Electron. Mater. 2009, 38, 1163. (12) Ferry, D. K.; Goodnick, S. M. Transport in Nanostructures; Cambridge University Press: Cambridge, 1997; pp 91208. (13) Landauer, R. IBM J. Res. Dev. 1957, 1, 223. (14) Assuming S is given approximately by eq 3 the assumptions eV , kT and ΔT , T are conservatively and self-consistently fulfilled when (Eres μ) e kT. (15) van Houten, H.; Beenakker, C. W. J.; Staring, A. A. M. In Single Charge Tuneling; Grabert, H., Devoret, M. H., Eds.; NATO ASI Series 294; Plenum Press: New York, 1992. (16) Bj€ork, M. T.; Thelander, C.; Hansen, A. E.; Jensen, L. E.; Larsson, M. W.; Wallenberg, L. R.; Samuelson, L. Nano Lett. 2004, 4, 1621. (17) Hoffmann, E. A.; Nilsson, H. A.; Matthews, J. E.; Nakpathomkun, N.; Persson, A. I.; Samuelson, L.; Linke, H. Nano Lett. 2009, 9, 779. (18) Shi, L.; Li, D. Y.; Yu, C. H.; Jang, W. Y.; Kim, D.; Yao, Z.; Kim, P.; Majumdar, A. ASME J. Heat Transfer 2003, 125, 881. (19) Molenkamp, L. W.; van Houten, H.; Staring, A. A. M.; Beenakker, C. W. J. Phys. Scr. 1993, T49b, 441. (20) Hirvi, K. P.; Kauppinen, J. P.; Korotkov, A. N.; Paalanen, M. A.; Pekola, J. P. Czech. J. Phys. 1996, 46, 3345. (21) Dubi, Y.; Di Ventra, M. Rev. Mod. Phys. 2011, 83, 131.
Corresponding Author
*E-mail:
[email protected]. Present Addresses †
)
Department of Mechanical Engineering, Oregon State University, Corvallis, OR 97331. ‡ Department of Physics, Thammasat University, Pathum Thani, Thailand, 12120. § Center for NanoScience and Fakult€at f€ur Physik, LudwigMaximilians-Universit€at, 80539 Munich, Germany. Division of Solid State Physics and The Nanometer Structure Consortium (nmC@LU), Lund University, Box 118, S-22100 Lund, Sweden.
’ ACKNOWLEDGMENT This research was sponsored by the ONR/ONAMI Nanometrology Initiative. ’ REFERENCES (1) Dresselhaus, M. S.; Chen, G.; Tang, M. Y.; Yang, R.; Lee, H.; Wang, D.; Ren, Z.; Fleurial, J.-P.; Gogna, P. Adv. Mater. 2007, 19, 1043. 4681
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