DE GRUYTER

Communications in Applied and Industrial Mathematics ISSN 2038-0909

Research Article

Commun. Appl. Ind. Math. 7 (2), 2016, 98-110

DOI: 10.1515/caim-2016-0008

Optimization of supercooling effect in nanoscaled thermoelectric layers

Abstract

In this paper we address the problem of optimization of the so called supercooling effect in thermoelectric nanoscaled layers. The effect arises when a short term electric pulse is applied to the layer. The analysis is based on constitutive equations of the Maxwell-Cattaneo type describing the time evolution of dissipative flows with the thermal and electric conductivities depending on the width of the layer. This introduces memory and nonlocal effects and consequently a wave-like behaviour of system's temperature. We study the effects of the shape of the electric pulse on the maximum diminishing of temperature by applying pulses of the form ta with a a power going from 0 to 10. Pulses with a a fractionary number perform better for nanoscaled devices whereas those with a bigger than unity do it for microscaled ones. We also find that the supercooling effect is improved by a factor of 6.6 over long length scale devices in the best performances and that the elapsed supercooling time for the nanoscaled devices equals the best of the microscaled ones. We use the spectral methods of solution which assure a well representation of wave behaviour of heat and electric charge in short time scales given their spectral convergence.

Keywords: Heat Transfer, Supercooling Effect, Nanoscaled Thermoelectric Layers.

AMS subject classification: 80A02. 1. Introduction

In this work we are interested in the problem of optimizing the shape of the electric pulse to produce the supercooling effect in thermoelectric devices, useful when it is needed an overcooling during a short time. Pulsed regimes produce a lower temperature than that obtained in the stationary state even with the optimal electric current for both uniform and nonuniform materials. This phenomenon is due to the fact that Peltier effect

© 2016 Ivan Rivera, Aldo Figueroa, Federico Vazquez, licensee De Gruyter Open.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.

Iván Rivera1, Aldo Figueroa2, Federico Vázquez2

1 Instituto de Investigación en Ciencias Básicas y Aplicadas. 2Centro de Investigación en Ciencias. Universidad Autónoma del Estado de Morelos, Cuernavaca, Morelos, Mexico,

*Email address for correspondence: vazquez@uaem.mx

Communicated by Vito Antonio Cimmelli and David Jou Received on December 30, 2014. Accepted on June 14, 2015.

occurs mainly at the cold junction while Joule heating is distributed in the bulk introducing a difference in the time taken by each one to influence the cold side of the device. The cold temperature is first changed by Peltier effect and after diffusion Joule heat reaches the cold junction affecting it. Some examples of devices which need to be overcooled during a short time are mid-IR laser gas sensors [1], condensation hygrometers and microelectronic processors generating hotspots [2], [3], [4], [5], [6]. The effect of the pulse form has been widely studied in macroscale of lengths. It has been shown that by applying a quadratic pulse form, the supercooling effect can be improved over other forms [7]. Some other pulse forms present additional advantages [8], [9]. Here we explore the influence of the electric pulse shape in the supercooling effect when the dimensions of the thermoelectric device goes to the submicrometer length scale.

Due to the progress in the microscaling techniques the thickness of thermoelectric materials may be reduced to become comparable to the phonon mean free path (PMFP) in the material. When the thickness is below the PMFP the heat and charged particles transport regime becomes of the wave type. We assume that the classical constitutive equations together with continuity and energy conservation equations represent a valid autonomous mesoscopic theory to describe heat and particle transfer in the system at such length scales. Few experimental analysis on transient Peltier cooling in pulsed operation are available at the submicrometric scales. In [10], by using thermoreflectance imaging microscopy, an experimental characterization of high-speed coplanar SiGe superlattice microcoolers subjected to repeated current pulses has been presented. The results show how complex the transport phenomena in on-chip Peltier coolers may be. They also make evident the necessity of modelling heat and electric charge transport in pulsed thermoelectricity in the sub-microscale of lengths. In the present investigation we theoretically examine the pulsed cooling of uniform thermoelectric materials in the submicrometric length scales. At these scales nonlocal and memory effects must be included, as well as size effects on the transport coefficients in order to describe adequately the known drop in the fluxes associated to the system's size reduction ( [11], [12], [13]) besides nonlinearities due to the presence of high temperature gradients ( [14]). The relationship between conductivities and size can be theoretically obtained from the higher order fluxes formulation of Extended Irreversible Thermodynamics (EIT) [11], [15] and it will be used here. We consider memory and nonlocal effects in the internal energy conservation equation through the use of Maxwell-Cattaneo (MC) type equations for the dissipa-tive fluxes. This introduces thermal and electric inertia in such a way that the resulting transport equation describes a wave-like behaviour of system's

temperature ( [16], [17]). The hyperbolic type transport equation obtained from the above procedure is here solved numerically for a thermoelectric thin film (see Figure 1) subjected to a Dirichlet boundary condition in the hot side of the thermoelectric device and a Robin type one in the cold side. The response of the system to a short pulse superimposed to the stationary state obtained with the optimal electric current is studied. Then the influence of the shape of the pulse in the thermal performance is investigated. The survey is as follows. In the next section we briefly expose the hyperbolic transport equation used to describe the time evolution of temperature (for details see [18]). In Section 3 our main results are displayed. A final section contains a discussion and concluding remarks.

Metal s Films"

Metal Conductors

Cold Substrate, Tc

p type

n type

Substrate, Heat Sink, 7h

Figure 1. Scheme of the thermoelectric device. This study is devoted to the thermal analysis of the thermoelectric material denoted by n type. The thickness (width) L goes from the micro to nanoscale of lengths.

2. Mathematical model

The heat transport in a single thermoelectric thin film has been modelled previously in [18]. In order to make self-contained this paper we include here a brief of the model. The temperature is described by a one-dimensional hyperbolic partial differential equation that in dimensionless terms reads (see Appendix A in reference [18])

d2T dT d2T ot2 fdT\

(1) f HOT + dT = + J + ^UJ'

where the first term the left hand side introduces the wave-like transport mechanism, whereas the second term is the attenuator of the wave. The

second term on the right hand side is the Joule heating and the third is the temporal variation of the power density due to the imposed electric field. The dimensionless coefficients in equation (1) are defined as

a Teff v K (L)T

aeff = — ' v = PCL2 '

(3) a = JoT z = JoTeff Se

( ) a pCpa(L)Th' z pCpL '

where pCp is the volumetric heat capacity, J0 is the magnitude of the electric current through the film, and Th is the fixed temperature at the hot side, K is the thermal conductivity, a is the electrical conductivity, SE is the Seebeck coefficient. The effective relaxation time Teff is given in terms of the collision time tc = lp/v, where lp is the mean free path of heat carriers and v denotes the average velocity of the heat carriers [11], that is Teff = Tc/4. Note that K is a function of the thickness. It is given by [18], [11]

(4) K <l> = Ko| (fMiy -

Similarly to the heat flux case, size effects on the electric charge transport can be considered by assuming along with EIT that the electric flux is an independent thermodynamic variable being described by a generalized constitutive equation. As a consequence, the electric conductivity a also depends on the thickness of the layer. We assume that this relationship has the form

(5) a(L) = 022 h/1 + 4^) - 1

2n2l2 IV \L

In expression (4), K0 the bulk thermal conductivity. The thermal conductivity K tends to the bulk value K0 when lp/L ^ 0, i.e. when the system's size is much greater than the mean free path of carriers. On the other hand, the equation (4) describes well the experimentally observed reduction of the thermal conductivity when L is in the order of magnitude of lp as it may be seen in [11]. In Eq. (5), le is the mean free path of electric charge carriers and ao the bulk electrical conductivity. As in the case of thermal conductivity, it predicts a reduction in the electrical conductivity

as the width if the layer is brought to the nanoscale. Some experimental evidence which support the postulate (Eq. (5)) was found in polycrystalline thin gold films [19]. The phonon's mean free path lp is related with resistive scattering and normal scattering of the phonons [14]. The relation is given by lp = c2rnrv/5 being c the average phonons speed, Tn the relaxation time of normal collisions and tp the relaxation time of phonons. On the other hand le, which is considerably smaller than lp, depends on the scattering of electrons as le = vFtp with vF the Fermi velocity.

We must note that in the calculation of the coefficient Z (Eq. 3), the bulk Seebeck coefficient was used. It is worth mentioning that if electron mean-free path is affected by the width of the thermoelectric, that would distort the electron distribution due to thermal gradients as well, so one would expect the Seebeck coefficient to be length dependent as well. The thermal and electrical conductivities relations with the length (Eqs. (4-5)) have an experimental basis. However, to our best knowledge, there is no information about the Seebeck coefficient dependency with the width. The latter would be an interesting aim for experimental research. The diffusive time scale is defined as follows

/a\ n2L2pCp

(6) T = ^KLT •

It should be noted that, when using the previous diffusive time scale T (Eq. (6)), the coefficient v (Eq. (3)) does not vary with L. It remains constant (v = n2/4). However, the a, ft and Z coefficients do vary with the length of the system. On the one hand, for L > 1 x 10-6 m, the coefficients aeff and Z are several orders of magnitude smaller than the remaining coefficients, that is, Eq. (1) takes the form of an elliptic equation for the thermoelectric cooler. On the other hand, for L < 1 x 10-7 m, the terms with the coefficients ae// and Z play an important role in the heat transport, see Table 1 in reference [18] for further information. Note that equation (1) is valid for times between the collision and the effective relaxation time, and lengths between the wave length of heat carriers and the diffusion length. For convenience in solving the equation (1), we choose the space domain as —1 < x < 1. Thus, the characteristic length is defined as l = L/2, where L is the system's size (thickness). The boundary conditions (in dimensionless variables) can be written as

<7> dx

= Y JT, T (1) = 1,

x= —1

where the dimensionless coefficient 7 is defined as 7 = SE J0l/K. Mention must be made that the dependency in length of SE is not important for the coefficient 7, since it is only present at the boundary condition, that is, it is a surface effect. The left boundary condition can be seen as a Robin-type since the temperature's gradient is proportional to the temperature. The right (Dirichlet) boundary condition denotes the constant value of temperature at the hot side. As the initial condition, we state that the device is at room temperature (hot side temperature), that is, T(x, 0) = 1. The hyperbolic equation (1) shows step-solutions which are very challenging numerically speaking [20]. Thus, it was solved using a numerical code based on a high order numerical scheme, namely, the Spectral Chebyshev Collocation method (SCC) [21], [22]. The time integration is explicit, whereas second order central differences schemes were used for the time derivatives.

The SCC method assumes that an unknown partial differential equation solution can be represented by a global, interpolating, Chebyshev partial sum. This finite series representing the solution is substituted into the differential equation and the coefficients determined so that the differential equation is satisfied at certain points within the range under consideration, which are defined following the Gauss-Lobatto collocation. The positions of the points in the range are chosen to make small the residual obtained when the approximate solution is substituted into the differential equation. In a finite-difference method, the approximation of a derivative at a grid point involves only very few neighbouring grid values of the function, while the Chebyshev approximation involves all the grid values. The global character of spectral methods is beneficial for accuracy. An additional property of the spectral methods is the easiness with which the accuracy of the computed solution can be estimated. In the steady state, with N = 30 collocation points, the maximum error between dimensionless analytic and numerical approximation is 1 x 10-11 [18].

3. Results

Since Silicon is a basic material for short and long scale devices, our results come from considering doped Silicon as working material, whose properties have been published before in [23]: K0 = 149 Wm-1K-1, a0 = 35.5 x 103 Q-1m-1, SE = 440 x 10-6 VK-1 and a = 88 x 10-6 m2s-1, where a is the thermal diffusivity; as hot side temperature, we take Th = 373.1 K. The mean free path of the heat carriers and their mean velocity are assumed to be [11] lp = 40 nm and v = 3K/pCplp, respectively. The mean free path of electric charge carriers was assumed to be of the order of the lattice constant of Silicon le = 0.5 nm.

In Figure 2, it can be seen the different shapes of the imposed electric pulse as a function of time. The duration in all cases is 0.163 and the maximum magnitude 3.5 over the stationary electric density (with normalized magnitude of one). The values are the optimal in order to obtain the maximum supercooling for the squared shape (t0), [17]. In the same figure, t0 denotes the squared shape pulse. In Figure 3 it can be seen the time evolution of the temperature at the cold side Tc of the thermoelectric for two distinct thickness of the film, namely, (a) L = 1 x 10-4 m and (b) L = 1 x 10-8 m. Each curve corresponds to one of the shaped pulses accordingly with the notation of Figure 2. The temperature evolves from the initial room temperature T0 =373.1 K to a steady cooling temperature Ts =369.9 K and 318.2 K for the micrometric and nanometric case, respectively, featuring a damped wave-like behaviour for the smaller device [18]. After the steady state is reached, the pulses are applied. Note that the change of temperature from the room temperature to the stationary temperature is not shown in Figures 3-4, which present an offset, setting t = 0 when the pulsed is started. Although several features can be observed in the transient stages, we remark the existence of a short term extra cooling (super cooling) in all cases. A zoom-in of the supercooling in the microscopic and nanoscopic scales during the applied electric pulse can be found in Figure 4(a) and (b), respectively. The curves in Figure 4(a) reproduces the previous result found in [7] (microscopic case). The curves in Figure 4(b) are the result of the present analysis. It is remarkable the fact that the super cooling effect is about 8 times larger at nanometric than at mi-crometric scale. Finally, in Table 1 a comparison of the supercooling and the effective supercooling area between the curve and the time axes for the microscopic and the nanoscopic scale is made. In spite this has no physical meaning, it is interesting the comparison. We discuss these results in the following section.

4. Discussion and Concluding Remarks

To begin with this last section, we remind briefly how the device has been operated. Once the optimal current is flowing through, the cold side shows a short smooth transient from the room temperature (Th = 373.1 K at t = 0) to a steady state temperature, thus obtaining a maximum cooling. In the steady state regime, a pulse of electric current J is applied at certain time and a short term extra cooling is achieved. The resulting behaviour is described by Eq.(1) which was solved using a spectral collocation scheme. In Eq.(1) the transport coefficients were taken as constant properties of the material in the temperature range considered in this study.

0 0.163

Figure 2. Electric current density J a a function of time. Pulse shapes.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 t

0 2 4 6 8 10 12 14 16

Figure 3. a) Cold side temperature TC as a function of time t for due to pulse shapes. a) Microscale. b) Nanoscale.

Figure 4. a) Zoom of Figure 3. Cold side temperature Tc as a function of time t for due to pulse shapes. a) Microscale. b) Nanoscale. Fractional pulse shapes perform better at the nanoscale.

Table 1. AT and effective cooling area for different pulse shapes.

Microscale

Nanoscale

Pulse AT [K] Area x10-4 [K] AT [K] Area x10-3 [K]

t0 1.09 1.98 7.62 2.83

¿Vio 1.05 2.36 7.88 2.92

t1/9 1.04 2.39 7.89 2.93

ti/s 1.03 2.44 7.90 2.94

t1/7 1.02 2.49 7.92 2.95

ti/6 1.02 2.56 7.93 2.95

t1/5 1.01 2.64 7.92 2.96

t1/4 0.99 2.76 7.88 2.95

t1/3 0.98 2.92 7.78 2.93

t1/2 0.96 3.06 7.45 2.82

ti 0.98 2.83 6.35 2.39

t2 1.14 2.13 4.71 1.74

t3 1.20 1.66 3.71 1.34

t4 1.20 1.35 3.07 1.09

t5 1.18 1.14 2.62 0.91

t6 1.15 0.98 2.28 0.79

t7 1.12 0.86 2.03 0.69

t8 1.10 0.76 1.82 0.62

t9 1.07 0.69 1.66 0.56

t10 1.04 0.63 1.52 0.51

In the case of the Seebeck coefficient this means an error of about 11 % while the electric conductivity coefficient introduces an error of about 6 %. The thermal conductivity is practically constant in the same range of temperatures. The transport model Eq.(1) incorporates thermal and electric conductivities depending on the thickness of the thermoelectric film. It also considers memory effects by assuming the Maxwell-Cattaneo form of the constitutive equations of heat and electric fluxes. We now discuss general features of the obtained results. The temperature T shows a minimum point, followed by a heating overshoot as can be easily seen in both the microscopic and the nanometric scales (Figure 3). The wave-like time evolution of temperature observed in Figure 3b) at the smallest length scale is a consequence of the thermal inertia of the system. This behavior comes from the time term introduced by the Maxwell-Cattaneo model used for the heat flux. In the same figure we can see two time spans where the su-

percooling effect is manifest. The first one occurs during the application of the pulse and the second one between t = 4 and t = 6. In Figure 4 we have included a zoom-in concerning the response of the system when the different pulses have been applied. The obtained extra cooling in the case of the microscopic length scale is ATp = Ts — Tp & 1 K when the temperature has reached a minimum Tp & 368.8 K. This value is comparable with that obtained analytically and numerically for homogeneous Silicon in [23] (Figure 2(b) in the cited reference). At the smaller scale, L = 1 x 10-8 m, the transient shows an oscillating decaying behaviour converging to a steady state temperature (Ts & 318 K at t & 16), see figure 4b). The application of the electric pulse produces an extra cooling of ATp & 8 K. It is worth mentioning that the elapsed supercooling time in the nanoscale is greater than in the microscale case. The supercooling time for the nanoscaled device is longer than the duration of the electric pulse by a factor of 1.5. In fact, all the minima are reached when the pulse has ceased at t = 0.163. Concerning the effects of the different pulse shapes, the results showed in Figure 4 indicate, on the one hand, that the pulses t3 and t6 improve the supercooling with ATp & 1.2 K in the micrometric lenth scale (Figure 4a)). On the other hand, in the nanometric case, the pulses t0, t1/6 and t1/3 show a ATp of about 8 K.

Most authors optimize the pulse shapes guided only by the deepest peak of the supercooling effect. This could be misleading, since cooling is not instantaneous, it is time dependent. With this in mind, the performance of each shape pulse can also be evaluated through the elapsed time by the supercooling effect. We measure it in an indirect way by calculating the area between the curve and the time axis. This is shown in Table 1 for shape pulses going from t0 to t10. In Table 1 it can also be seen the maximum extra cooling ATp reached during the supercooling effect. Clearly, in the microscopic length scale the value ATp & 1.2 K for t3 is a maximum. In the nanoscopic case the value ATp & 7.93 K for t1/6 is also a maximum. However, seeing from another point of view, the supercooling elapsed time shows that the best pulse shapes are t1/2 and t1/5 for the microscale and nanoscale, respectively.

In the microscale, results in Table 1 agree with [7], by only considering the supercooling peak in the optimization process, where pulse shapes due to whole number powers perform better. However, the latter could be misleading, since by considering the supercooling elapsed time, fractional powers for the pulse shape show wider effective supercooling area.

Finally we summarize our results. First, the transient for the nanomet-ric layer shows an oscillating decaying behaviour towards the stationary state, in contradistinction with the microscopic case which shows a smooth

decaying to the stationary state. Second, pulses with a a fractionary number perform better for nanoscaled devices whereas those with a bigger than unity do it for microscaled ones. Third, the supercooling effect is improved by a factor of 6.6 over long length scale devices in the best performances and fourth, the maximum elapsed supercooling time for the nanoscaled devices equals that of the microscaled ones.

Acknowledgements.

A. Figueroa thanks a postdoctoral fellowship and the Catedras program from CONACYT. F. Vazquez acknowledges financial support from PROMEP and CONACYT (Mexico) under grant 133763.

REFERENCES

1. G. J. Snyder, J. P. Fleurial, T. Caillat, R. Yang, and G. Chen, Supercooling of peltier cooler using a current pulse, Journal of Applied Physics, vol. 92, no. 1564, 2002.

2. M. P. Gupta, M. Sayer, S. Mukhopadhyay, and S. Kumar, On-chip peltier cooling using current pulse, in 12th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems (ITherm), pp. 1-7, 2010.

3. M. P. Gupta, M.-H. Sayer, S. Mukhopadhyay, and S. Kumar, Ultrathin thermoelectric devices for on-chip peltier cooling, in IEEE Transactions on Components, Packaging and Manufacturing Technology, vol. 1, pp. 1395-1405, 2011.

4. B. Alexandrov, O. S. S. Kumar, and S. Mukhopadhyay, Prospects of active cooling with integrated super-lattice based thin-film thermoelectric devices for mitigating hotspot challenges in microprocessors, in 17th Asia and South Pacific Design Automation Conference (ASP-DAC), pp. 633-638, 2012.

5. M. Manno, P. Wang, and A. Bar-Cohen, Anticipatory thermoelectric cooling of a transient germanium hotspot, in ASME 2013 International Technical Conference and Exhibition on Packaging and Integration of Electronic and Photonic Microsystems, vol. 2, p. V002T08A038, 2013.

6. I. Chowdhury, R. Prasher, K. Lofgreen, G. Chrysler, S. Narasimhan, R. Mahajan, D. Koester, R. Alley, and R. Venkatasubramanian, On-chip cooling by superlattice-based thin-film thermoelectrics, Nature Nan-otechnology, vol. 4, pp. 235-238, 2009.

7. T. Thonhauser, G. D. Mahan, L. Zikatanov, and J. Roe, Improved supercooling in transient thermoelectrics, Applied Physics Letters, vol. 85, no. 15, pp. 3247-3249, 2004.

8. J. N. Mao, H. X. Chen, H. Jia, and X. L. Qian, The transient be-

haviour of peltier junctions pulsed with supercooling, Journal of Applied Physics, vol. 112, no. 014514, 2012.

9. M. Ma and J. Yu, A numerical study on the temperature overshoot characteristic of a realistic thermoelectric module under current pulse operation, International Journal of Heat and Mass Transfer, vol. 72, pp. 234-241, 2014.

10. B. Vermeersch, J.-H. Bahk, J. Christofferson, and A. Shakouri, Ther-moreflectance imaging of sub 100ns pulsed cooling in high-speed thermoelectric microcoolers, Journal of Applied Physics, vol. 113, no. 104502,

11. F. X. Alvarez and D. Jou, Memory and nonlocal effects in heat transport: From diffusive to ballistic regimes, Applied Physics Letters, vol. 90, pp. 1-3, 2007.

12. A. Sellitto, V. Cimmelli, and D. Jou, Thermoelectric effects and size dependency of the figure-of-merit in cylindrical nanowires, International Journal of Heat and Mass Transfer, vol. 57, pp. 109-116, 2013.

13. F. Márkus and K. Gambár, Heat propagation dynamics in thin silicon layers, International Journal of Heat and Mass Transfer, vol. 56, pp. 495-500, 2013.

14. A. Sellitto, V. Cimmelli, and D. Jou, Analysis of three nonlinear effects in a continuum approach to heat transport in nanosystems, Physica D, vol. 241, pp. 1344-1350, 2012.

15. D. Jou, J. Casas-Vazquez, and G. Lebon, Extended Irreversible Thermodynamics. Springer, 2010.

16. A. Sellitto and V. Cimmelli, A continuum approach to thermomass theory, Journal of Heat Transfer, vol. 134, no. 112402, 2012.

17. F. Vazquez and A. Figueroa, Pulsed cooling in nanoscaled thermoelectric layers: nonlocal and memory effects, Submitted, 2014.

18. A. Figueroa and F. Vázquez, Optimal performance and entropy generation transition from micro to nanoscaled thermoelectric layers, International Journal of Heat and Mass Transfer, vol. 71, pp. 724-731,

19. W. G. Ma, H. D. Wang, X. Zhang, and W. Wang, Experiment study of the size effects on electron-phonon relaxation and electrical resistivity of polycrystalline thin gold films, Journal of Applied Physics, vol. 108, no. 064308, 2010.

20. A. Figueroa and F. Vazquez, Spectral and finite difference solutions of the hyperbolic heat transport equation for thermoelectric thin films, Applied Mathematics, vol. 4, pp. 22-27, 2013.

21. S. Zhao and M. J. Yedlin, A new iterative chebyshev spectral method for solving the elliptic equation V ■ (aVu) = f, Journal of Computational

Physics, vol. 113, pp. 251-223, 1994.

22. R. Peyret, Spectral Methods for Incompressible Viscous Flow. SpringerVerlag, 2002.

23. Q. Zhou, Z. Bian, and A. Shakouri, Pulsed cooling of inhomogeneous thermoelectric materials, Journal of Physics D: Applied Physics, vol. 40, no. 064308, pp. 4376-4381, 2007.