The inverse coefficient problem of heat transfer in layered nanostructures

The rapid development of electronics leads to the creation and use of electronic components of small dimensions, including nanoelements of complex, layered structure. The search for effective methods for cooling electronic systems dictates the need for the development of methods for the numerical analysis of heat transfer in nanostructures. A characteristic feature of energy transfer in such systems is the dominant role of contact thermal resistance at interlayer interfaces. Since the contact resistance depends on a number of factors associated with the technology of heterostructures manufacturing, it is of great importance to determine the corresponding coefficients from the results of temperature measurements.
The purpose of this paper is to evaluate the possibility of reconstructing the thermal resistance coefficients at the interfaces between layers by solving the inverse problem of heat transfer.
The complex of algorithms includes two major blocks — a block for solving the direct heat transfer problem in a layered nanostructure and an optimization block for solving the inverse problem. The direct problem was formulated in an algebraic (finite difference) form under the assumption of a constant temperature within each layer, which is due to the small thickness of the layers. The inverse problem was solved in the extreme formulation, the optimization was carried out using zero-order methods that do not require the calculation of the derivatives of the optimized function. As a basic optimization algorithm, the Nelder—Mead method was used in combination with random restarts to search for a global minimum.
The results of the identification of the contact thermal resistance coefficients obtained in the framework of a quasi-real experiment are presented. The accuracy of the identification problem solution is estimated as a function of the number of layers in the heterostructure and the «measurements» error.
The obtained results are planned to be used in the new technique of multiscale modeling of thermal regimes of the electronic component base of the microwave range, when identifying the coefficients of thermal conductivity of heterostructure.

heat transfer, layered nanostructure, heterostructure, contact thermal resistance coefficients, interface, inverse problem

1. Borisenko V. E., Vorob’eva A. I., Utkina E.A. Nanoelektronika [Nanoelectronics]. Moscow: Binom. Laboratoriya znanii, 2009, 223 p. (In Russ.)

2. Vasileska D., Goodnick S. M., Goodnick S. Computational electronics: semiclassical and quantum device modeling and simulation. CRC Press, 2010, 782 p.

3. Chu R. C. The challenging of electronic cooling: past, current and future. J. Electron. Packag, 2004, vol. 126, no. 4, pp. 491—500. DOI: 10.1115/1.1839594

4. Dudinov K. V., Ippolitov V. M., Klimova A. V., Pashkovsky A. B., Samsonova I. V. Features of heat release in high-power field-effect transistors. Radiotekhnika = Radioengineering, 2007, no. 3, pp. 60—62. (In Russ.)

5. Berezhnova P. V., Pashkovsky A. B., Ratnikova A. K., Lukashin V. M. Valuation of non-local heat generation area in power field-effect transistors on heterostructures. Electronnaya Tekhnika. Series 1: SVCH-Tekhnika = Electronic Engineering. Ser. 1: Microwave Engineering, 2007, no. 4, pp. 21—24. (In Russ.)

6. Protasov D. Y., Malin T. V., Tikhonov A. V., Zhuravlev K. S., Tsatsulnikov A. F. Electron scattering in AlGaN/GaN heterostructures with a two-dimensional electron gas. Semiconductors, 2013, vol. 47, no. 1, pp. 33—44. DOI: 10.1134/S1063782613010181

7. Abgaryan K. K., Reviznikov D. L. Numerical simulation of the distribution of charge carrier in nanosized semiconductor heterostructures with account for polarization effects. Computational Mathematics and Mathematical Physics, 2016, vol. 56, no. 1, pp. 161—172. DOI: 10.1134/S0965542516010048

8. Abgaryan K. K., Mutigullin I. V., Reviznikov D. L. Computational model of 2DEG mobility in the AlGaN/GaN heterostructures. Phys. status solidi (c), 2015, vol. 12, no. 4-5, pp. 460—465. DOI: 10.1002/pssc.201400200

9. Dmitriev A. S. Vvedenie v nanoteplofiziku [Introduction to nano-thermal physics]. Moscow: Binom. Laboratoriya znanii, 2015, 792 p. (In Russ.)

10. Khvesyuk V. I. Heat distribution in multilayer nanostructures. Pis’ma v zhurnal tekhnicheskoi fiziki = Technical Physics Letters, 2016, vol. 42, no. 19, pp. 20—25. (In Russ.)

11. Khvesyuk V. I., Skryabin A. S. Heat conduction in nanostructures. High Temperature, 2017, vol. 55, no. 3, pp. 434—456. DOI: 10.1134/S0018151X17030129

12. Cahill D. G., Ford W. K., Goodson K. E., Mahan G. D., Majumdar A., Maris H. J., Merlin R., Phillpot S. R. Nanoscale thermal transport. J. Appl. Phys., 2003, vol. 93, no. 2, pp. 793—818. DOI: 10.1063/1.1524305

13. Chen G. Nanoscale Energy Transport and Conversion:

14. A Parallel Treatment of Electrons, Molecules, Phonons, and Photons. Oxford University Press, 2005, 560 p.

15. Termentzidis K., Parasuraman J., Cruz C. A. D., Merabia S., Angelescu D., Marty F., Bourouina T., X. Kleber, Chantrenne P., Basset P. Thermal conductivity and thermal boundary resistance of nanostructures. Nanoscale Res. Lett., 2011, vol. 6, p. 288 (10 pp.). DOI: 10.1186/1556-276X-6-288

16. Madhusudana C. V. Thermal contact conductance. New York: Springer-Verlag, 1996, 168 p. DOI: 10.1007/978-1-4612-3978-9

17. Samvedi V., Tomar V. The role of interface thermal boundary resistance in the overall thermal conductivity of Si–Ge multilayered structures. Nanotechnology, 2009, vol. 20, no. 36, art. 365701. DOI: 10.1088/0957-4484/20/36/365701

18. Samarsky A. A., Vabishchevich P. N. Chislennye metody resheniya obratnykh zadach matematicheskoi fiziki [Numerical methods for solving inverse problems of mathematical physics]. Moscow: Editorial URSS, 2004, 480 p. (In Russ.)

19. Alifanov O. M. Obratnye zadachi teploobmena [Inverse problems of heat transfer]. Moscow: Mashinostroenie, 1988, 280 p. (In Russ.)

20. Abgaryan K. K. Optimization problems of nanoscale semiconductor heterostructures. Izvestiya Vysshikh Uchebnykh Zavedenii. Materialy Elektronnoi Tekhniki = Materials of Electronics Engineering, 2016, vol. 19, no. 2, pp. 108—114. (In Russ.). DOI: 10.17073/1609-3577-2016-2-108-114

21. Abgaryan K. K., Reviznikov D. L. Vychislitel‘nye algoritmy v zadachakh modelirovaniya i optimizatsii poluprovodnikovykh geterostruktur [Computational algorithms in problems of modeling and optimization of semiconductor heterostructures]. Moscow: MAKS Press, 2016, 120 p. (In Russ.)

22. Vorob’ev D. A., Hvesyuk V. I. Calculation method for non-stationary heating of nano-structures. Science and Education of Bauman MSTU, 2013, pp. 541—550. (In Russ.). DOI: 10.7463/0913.0617255

23. Sadao Adachi. Properties of Semiconductors Alloys: Group-IV, III-V and II-VI Semiconductors. John Wiley & Sons, 2009, 422 p. DOI: 10.1002/9780470744383