Asymmetry in particle transport in slightly non-homogeneously doped silicon layers in low injection regime and quasi-neutrality condition


  • Víctor Hernández Universidad Autónoma del Estado de Morelos
  • Federico Vázquez Universidad Autónoma del Estado de Morelos



doped silicon films, inhomogeneous doping, low/high particle injection regime, irreversible thermodynamics, Shokley-Read-Hall recombination


In this work a 1D slightly doped n-type Silicon layer is considered. Irreversible thermodynamics transport equations are used to obtain the spatial particle distribution in isothermal stationary state. The excess particle transport is studied in low injection regime and quasineutrality condition. The material is subjected to Dirichlet-Neumann (N-D) conditions at the boundaries. We wonder if an asymmetry of the particle flux exists when the boundary conditions are inverted. We find that the asymmetry does exist and a rectification factor may reach the value 0.35. We conclude that particle flux rectification seems to be featuring the particle transport in slightly non-homogeneously doped semiconductor when the excess hole concentration is smaller than the equilibrium hole concentration

Author Biography

Víctor Hernández, Universidad Autónoma del Estado de Morelos

Physics Department.


S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, 1962).

J.E. Parrott. Transport theory of semiconductor energy conversion, Journal of Applied Physics 53 (1982) 9105.

S. Selberherr, Analysis and Simulation of Semiconductor Devices (Springer, Berlin, 1984).

Gerhard K. Wachutka. Rigorous Thermodynamic Treatment of Heat Generation and Conduction in Semiconductor Device Modeling, IEEE Transactions on Computer-Aided Design, 9 (1990) 1141-1149.

O. Muscato. The Onsager reciprocity principle as a check of consistency for semiconductor carrier transport models, Physica A 289 (2001) 422-458.

H. Ch. Oettinger, Beyond Equilibrium Thermodynamics (Wiley, 2005).

S. Kjelstrup and D. Bedeaux, Non-equilibrium thermodynamics of heterogeneous systems (World Scientific 2008).

G. Lebon, D. Jou and J. Casas-Vázquez, Understanding Non-equilibrium Thermodynamics. Foundations, Applications, Frontiers (Springer-Verlag, Berlin-Heidelberg, 2008).

D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics, 4th Ed. (Springer, Berlin, 2010).

P. Ván, T. Fülöp. Universality in heat conduction theory: weakly nonlocal thermodynamics, Ann. Phys. (Berlin) 524 (2012) 470–478.

V. A. Cimmelli, D. Jou, T. Ruggeri, P. Ván, Entropy Principle and Recent Results in Non-Equilibrium Theories, Entropy 16 (2014) 1756-1807.

S. L. Sobolev, Effective temperature in nonequilibrium state with heat flux using discrete variable model, Physics Letters A 381 (2017) 2893-2897.

I. N. Volovichev, Yu. G. Gurevich, Generation–Recombination Processes in Semiconductors, Semiconductors 35 (2001) 306–315.

S. R. in ’t Hout, Quasineutrality in semiconductors, Journal of Applied Physics 79 (1996) 8435.

Z.-G. Chen, X. Shi, L.-D. Zhao, J. Zou, High performance SnSe thermo-electric mmaterials: Progress and future challenge. Progress in Materials Science 97 (2018) 283-346.

K. Imasato, S.D. Kang, S. Ohno, G.J. Snyder, Band engineering in Mg3Sb2 by alloying with Mg3Bi2 for enhanced thermoelectric performance, Mater. Horiz. 5 (2018) 59-64.

H. J. Goldsmid, Bismuth Telluride and Its Alloys as Materials for Thermoelec- tric Generation, Materials 7 (2014) 2577-2592.

A. Sankhla, H. Kamila, K. Kelm, E. Mueller and J. de Boor, Analyzing thermo- electric transport in n-type Mg2Si0.4Sn0.6 and correlation with microstructural effects: An insight on the role of Mg, Acta Materialia 199 (2020) 85-95.

H, Kamila, P. Sahu, A. Sankhla, M. Yasseri, H. N. Pham, T. Dasgupta, E. Mueller J. de Boor, Analyzing transport properties of p-type Mg2Si-Mg2Sn solid solutions: optimization of thermoelectric performance and insight into the electronic band structure, Journal of Materials Chemistry A 7 (2019) 1045.

B.L. Pedersen, H. Birkedal, B.B. Iversen, M. Nygren, P.T. Frederiksen, Influence of sample compaction on the thermoelectric performance of Zn4Sb3. Appl. Phys. Lett. 89 (2006) 242108.

Y. Wang, X. Zhang, Y. Liu, Y. Wang, J. Zhang and M. Yue, Optimizing the thermoelectric performance of p-type Mg3Sb2 by Sn doping, Vacuum 177 (2020) 109388.

[33] G. K. Goyal, S Mukherjee, R. C. Mallik, S. Vitta, I. Samajdar and T. Dasgupta, High Thermoelectric Performance in Mg2(Si0.3Sn0.7) by Enhanced Phonon Scattering, ACS Applied Energy Materials 2 (2019) 2129-2137.

N. Farahi, S. Prabhudev, G. A. Botton, J. R. Salvador and H. Kleinke, Nano- and Microstructure Engineering: An Effective Method for Creating High Effi- ciency Magnesium Silicide Based Thermoelectrics, ACS Appl. Mater. Interfaces 8 (2016) 34431-34437.

G. N. Isachenko, A Yu. Samunin, E. A. Gurieva, M. I. Federov, D. A. Pshenay- Severin, P. P. Konstantinov and M. D. Kamolova, Thermoelectric Properties of Nanostructured pMg2SixSn1−x (x = 0.2 to 0.4) Solid Solutions, Journal of Electronic Materials 45 (2016) 1982-1986.

G. Bernard-Granger, C. Navone, J. Laforestier, M. Boidot, K. Romanjek, J. Carrete and J. Simon, Microstructure investigations and thermoelectrical prop- erties of an N-type magnesium-silicon-tin alloy sintered from a gas-phase atomized powder, Acta Materialia 96 (2015) 437-451.

W. Shockley. The Theory of p-n Junctions in Semiconductors and p-n Junction Transistors, Bell System Technical Journal 28 (1949) 435-489.

W. Shockley and W. T. Read, Jr. Statistics of the Recombinations of Holes and Electrons, Physical Review 87 (1952) 835.

I. N. Volovichev and Yu. G. Gurevich. GenerationRecombination Processes in Semiconductors, Semiconductors 35 (2001) 306–315.

Yu. G. Gurevich, J. E. Velázquez-Pérez, G. Espejo-López, I. N. Volovichev, and O. Yu. Titov. Transport of nonequilibrium carriers in bipolar semiconductors, Journal of Applied Physics 101 (2007) 023705 .

B. El Filali, O. Yu Titov, Yu G. Gurevich. Physics of charge transport in metal–monopolar (n- or p-type) semiconductor–metal structures. Journal of Physics and Chemistry of Solids 118 (2018) 14-20.

Brahim El Filali, O. Yu. Titov, Ch. Ballardo Rodríguez, Yu. G. Gurevich, Heating and nonlinear charge transport in bipolar semiconductors, Physica Status Solidi B, May 2022,

A. N. Ishaque, J. W. Howard, M. Becker, and R. C. Block. An extended ambipolar model: Formulation, analytical investigations, and application to photocurrent modeling, Journal of Applied Physics 69 (1991) 307.

Donald A. Naemen, Semiconductor physics and devices basic principles (McGraw-Hill, N.Y., 2001).

Daniel Macdonald, Andrés Cuevas, Validity of simplified Shockley-Read-Hall statistics for modeling carrier lifetimes in crystalline silicon, Physical Review B 67 (2003) 075203.




How to Cite

V. . Hernández and F. Vázquez, “Asymmetry in particle transport in slightly non-homogeneously doped silicon layers in low injection regime and quasi-neutrality condition”, Rev. Mex. Fís., vol. 69, no. 1 Jan-Feb, pp. 011702 1–, Jan. 2023.



17 Thermodynamics and Statistical Physics