Charge Trapping and Variability in CMOS Technologiesat Cryogenic Temperatures

A.2 Saddlepoint Method

The stationary phase approximation or saddlepoint method was developed in the 18\( ^\mathrm {th} \) century by Pierre-Simon Laplace [252]. The aim is to find an approximation for the integral

(A.12) \{begin}{align} I=\int \limits _{-\infty }^\infty \mathrm {e}^{-f(x)}\mathrm {d}x \{end}{align}

for analytic functions \( f(x) \) with a global minimum at \( x_0 \), which implies that

(A.13) \{begin}{align} f’(x_0)=0 \{end}{align}

and

(A.14) \{begin}{align} f”(x_0)>0 \{end}{align}

holds. Expanding \( f(x) \) in a Taylor series at the minimum \( x_0 \) and simplifying the expression using \( f’(x_0)=0 \)

(A.15) \{begin}{align} \begin {split} f(x)&\approx f(x_0)+f’(x_0)(x-x_0)+\frac {1}{2}f”(x_0)(x-x_0)^2 \\ &= f(x_0)+\frac {1}{2}f”(x_0)(x-x_0)^2, \end {split} \{end}{align}

we get

(A.16) \{begin}{align} \begin {split} I&=\int \limits _{-\infty }^\infty \mathrm {e}^{-f(x)}\mathrm {d}x\\ &\approx \int \limits _{-\infty }^\infty \mathrm {e}^{-f(x_0)-\frac {1}{2}f”(x_0)(x-x_0)^2}\mathrm {d}x\\ &=\mathrm {e}^{-f(x_0)}\int \limits _{-\infty }^\infty \mathrm {e}^{-\frac {1}{2}f”(x_0)(x-x_0)^2}\mathrm {d}x. \label {eq:SPM_int_2} \end {split} \{end}{align}

Using the solution of the Gaussian integral

(A.17) \{begin}{align} \int \limits _{-\infty }^\infty \mathrm {e}^{-a(x+b)^2}\mathrm {d}x=\sqrt {\frac {\pi }{a}} \{end}{align}

we can evaluate (A.16) to get the final result

(A.18) \{begin}{align} I=\mathrm {e}^{-f(x_0)}\sqrt {\frac {2\pi }{f”(x_0)}}. \{end}{align}

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  • [248] B. Kaczer, T. Grasser, P. J. Roussel, J. Martin-Martinez, R. O’Connor, B. J. O’Sullivan, and G. Groeseneken. “Ubiquitous Relaxation in BTI Stressing-New Evaluation and Insights”. In: Proceedings of the International Reliability Physics Symposium (IRPS). 2008, pp. 20–27.

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List of Publications

Scientific Journals, Book Contribution

  • [MJJ1] J. Michl, A. Grill, D. Waldhör, W. Goes, B. Kaczer, D. Linten, B. Parvais, B. Govoreanu, I. Radu, M. Waltl, and T. Grasser. “Efficient Modeling of Charge Trapping at Cryogenic Temperatures-Part I: Theory”. In: IEEE Transactions on Electron Devices 68.12 (2021), pp. 6365–6371. DOI: 10.1109/TED.2021.3116931.

  • [MJJ2] D. Waldhör, C. Schleich, J. Michl, B. Stampfer, K. Tselios, E. Ioannidis, H. Enichlmair, M. Waltl, and T. Grasser. “Toward Automated Defect Extraction From Bias Temperature Instability Measurements”. In: IEEE Transactions on Electron Devices 68.8 (2021), pp. 4057–4063. DOI: 10.1109/TED.2021.3091966.

  • [MJJ3] K. Tselios, D. Waldhör, B. Stampfer, J. Michl, E. Ioannidis, H. Enichlmair, T. Grasser, and M. Waltl. “On the Distribution of Single Defect Threshold Voltage Shifts in SiON Transistors”. In: IEEE Transactions on Device and Materials Reliability 91.2 (2021), pp. 199–206. DOI: 10.1109/TDMR.2021.3080983.

  • [MJJ4] A. Beckers, J. Michl, A. Grill, B. Kaczer, M. G. Bardon, B. Parvais, B. Govoreanu, K. De Greve, G. Hiblot, and G. Hellings. “Physics-Based and Closed-Form Model for Cryogenic MOSFET Subthreshold Swing”. In: IEEE Transactions on Electron Devices (2022). Submitted.

  • [MJJ5] J. Michl, A. Grill, D. Waldhör, W. Goes, B. Kaczer, D. Linten, B. Parvais, B. Govoreanu, I. Radu, T. Grasser, and M. Waltl. “Efficient Modeling of Charge Trapping at Cryogenic Temperatures-Part II: Experimental”. In: IEEE Transactions on Electron Devices 68.12 (2021), pp. 6372–6378. DOI: 10.1109/TED.2021.3117740.

  • [MJJ6] C. Schleich, D. Waldhör, T. Knobloch, W. Zhou, B. Stampfer, J. Michl, M. Waltl, and T. Grasser. “Single-Versus Multi-Step Trap Assisted Tunneling Currents – Part I: Theory”. In: IEEE Transactions on Electron Devices (2022), pp. 1–7. DOI: 10.1109/TED.2022.3185966.

Conference Proceedings

  • [MJC1] K. Tselios, B. Stampfer, J. Michl, E. Ioannidis, H. Enichlmair, and M. Waltl. “Distribution of Step Heights of Electron and Hole Traps in SiON nMOS Transistors”. In: Proceedings of the International Integrated Reliability Workshop (IIRW). 2020, pp. 1–6. DOI: 10.1109/IIRW49815.2020.9312871.

  • [MJC2] K. Tselios, J. Michl, T. Knobloch, H. Enichlmair, E. Ioannidis, R. Minixhofer, T. Grasser, and M. Waltl. “Evaluation of the Impact of Defects on Threshold Voltage Drift Employing SiO2 pMOS Transistors”. In: Abstracts of the 26th European Symposium on Reliability of Electron Devices, Failure Physics and Analysis. Accepted. 2022, p. 99.

  • [MJC3] A. Grill, E. Bury, J. Michl, S. E. Tyaginov, D. Linten, T. Grasser, B. Parvais, B. Kaczer, M. Waltl, and I. Radu. “Reliability and Variability of Advanced CMOS Devices at Cryogenic Temperatures”. In: Proceedings of the IEEE International Reliability Physics Symposium (IRPS). 2020, pp. 1–6. ISBN: 978-1-7281-3199-3. DOI: 10.1109/IRPS45951.2020.9128316.

  • [MJC4] J. Michl, A. Grill, B. Stampfer, D. Waldhoer, C. Schleich, T. Knobloch, E. Ioannidis, H. Enichlmair, R. Minixhofer, B. Kaczer, B. Parvais, B. Govoreanu, I. Radu, T. Grasser, and M. Waltl. “Evidence of Tunneling Driven Random Telegraph Noise in Cryo-CMOS”. In: 2021 IEEE International Electron Devices Meeting (IEDM). 2021, pp. 31.3.1–31.3.4. DOI: 10.1109/IEDM19574.2021.9720501.

  • [MJC5] A. Grill, V. John, J. Michl, A. Beckers, E. Bury, S. Tyaginov, B. Parvais, A. V. Chasin, T. Grasser, M. Waltl, B. Kaczer, and B. Govoreanu. “Temperature Dependent Mismatch and Variability in a Cryo-CMOS Array with 30k Transistors”. In: 2022 IEEE International Reliability Physics Symposium (IRPS). 2022, 10A.1-1-10A.1–6. DOI: 10.1109/IRPS48227.2022.9764594.

  • [MJC6] T. Knobloch, J. Michl, D. Waldhör, Y. Illarionov, B. Stampfer, A. Grill, R. Zhou, P. Wu, M. Waltl, J. Appenzeller, and T. Grasser. “Analysis of Single Electron Traps in Nano-scaled MoS2 FETs at Cryogenic Temperatures”. In: Proceedings of the Device Research Conference (DRC). 2020, pp. 52–53.

  • [MJC7] J. Michl, A. Grill, D. Claes, G. Rzepa, B. Kaczer, D. Linten, I. Radu, T. Grasser, and M. Waltl. “Quantum Mechanical Charge Trap Modeling to Explain BTI at Cryogenic Temperatures”. In: Proceedings of the IEEE International Reliability Physics Symposium (IRPS). 2020, pp. 1–6. ISBN: 978-1-7281-3199-3. DOI: 10.1109/IRPS45951.2020.9128349.