LSSC 2019: Large-Scale Scientific Computing pp 403-410

# A Revised Wigner Function Approach for Stationary Quantum Transport

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11958)

## Abstract

The Wigner equation describing stationary quantum transport has a singularity at the point $$k=0$$. Deterministic solution methods usually deal with the singularity by just avoiding that point in the mesh (e.g., Frensley’s method). Results from such methods are known to depend strongly on the discretization and meshing parameters.

We propose a revised approach which explicitly includes the point $$k=0$$ in the mesh. For this we give two equations for $$k=0$$. The first condition is an algebraic constraint which ensures that the solution of the Wigner equation has no singularity for $$k=0$$. If this condition is fulfilled we then can derive a transport equation for $$k=0$$ as a secondary equation.

The resulting system with two equations for $$k=0$$ is overdetermined and we call it the constrained Wigner equation. We give a theoretical analysis of the overdeterminacy by relating the two equations for $$k=0$$ to boundary conditions for the sigma equation, which is the inverse Fourier transform of the Wigner equation.

We show results from a prototype implementation of the constrained equation which gives good agreement with results from the quantum transmitting boundary method. No numerical parameter fitting is needed.

## Keywords

Wigner function Sigma function Finite difference method Constrained equation Quantum transport Device simulation Resonant tunneling diode

## References

1. 1.
Arnold, A., Lange, H., Zweifel, P.F.: A discrete-velocity, stationary Wigner equation. J. Math. Phys. 41(11), 7167–7180 (2000).
2. 2.
Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004).
3. 3.
Frensley, W.R.: Boundary conditions for open quantum systems driven far from equilibrium. Rev. Mod. Phys. 62(3), 745–791 (1990).
4. 4.
Lent, C.S., Kirkner, D.: The quantum transmitting boundary method. J. Appl. Phys. 67(10), 6353–6359 (1990).
5. 5.
Li, R., Lu, T., Sun, Z.: Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions. Front. Math. China 12(4), 907–919 (2017).
6. 6.
Suryanarayana, M.B.: On multidimensional integral equations of Volterra type. Pac. J. Math. 41(3), 809–828 (1972).
7. 7.
Tsuchiya, H., Ogawa, M., Miyoshi, T.: Simulation of quantum transport in quantum devices with spatially varying effective mass. IEEE Trans. Electron Devices 38(6), 1246–1252 (1991).
8. 8.
Wigner, E.P.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40(5), 749–759 (1932).

© Springer Nature Switzerland AG 2020

## Authors and Affiliations

• Robert Kosik
• 1
Email author
• Johann Cervenka
• 1
• Mischa Thesberg
• 1
• Hans Kosina
• 1
1. 1.Institute for MicroelectronicsTU WienViennaAustria
{"ft_lang_label":"__label__en","ft_lang_prob":0.71799934,"math_prob":0.99190503,"size":3252,"snap":"2020-10-2020-16","text_gpt3_token_len":848,"char_repetition_ratio":0.13485222,"word_repetition_ratio":0.0,"special_character_ratio":0.28228784,"punctuation_ratio":0.23704867,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9953629,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-19T02:32:43Z\",\"WARC-Record-ID\":\"<urn:uuid:bd4d9ed1-0348-4642-92b3-3cc0b95f5b48>\",\"Content-Length\":\"77406\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:983a1f1e-1067-4f9a-8507-9c4b5b062d76>\",\"WARC-Concurrent-To\":\"<urn:uuid:dccd73a4-041e-4c87-8107-3e7edd5a54c0>\",\"WARC-IP-Address\":\"151.101.248.95\",\"WARC-Target-URI\":\"https://rd.springer.com/chapter/10.1007%2F978-3-030-41032-2_46\",\"WARC-Payload-Digest\":\"sha1:3APISPHDL6FTW4OY2PV5VLO3CDNVOCJ5\",\"WARC-Block-Digest\":\"sha1:VUX5WWIIIC56B226EADIVKPWQNB3UO47\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875143963.79_warc_CC-MAIN-20200219000604-20200219030604-00083.warc.gz\"}"}