Determination of doping and temperature-dependent elastic constants of degenerately doped silicon from MEMS resonators

Antti Jaakkola, Mika Prunnila, Tuomas Pensala, James R Dekker, Panu Pekko

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Abstract

Elastic constants c11, c12, and c44 of degenerately doped silicon are studied experimentally as a function of the doping level and temperature. First-and second-order temperature coefficients of the elastic constants are extracted from measured resonance frequencies of a set of MEMS resonators fabricated on seven different wafers doped with phosphorus (carrier concentrations 4.1, 4.7, and 7.5 x 1019 cm -3), arsenic (1.7 and 2.5 x 1019 cm-3), or boron (0.6 and 3 * 1019 cm-3). Measurements cover a temperature range from-40°C to +85°C. It is found that the linear temperature coefficient of the shear elastic parameter c11- c>sub>12 is zero at n-type doping level of n ~ 2 x 1019 cm -3, and that it increases to more than 40 ppm/K with increasing doping. This observation implies that the frequency of many types of resonance modes, including extensional bulk modes and flexural modes, can be temperature compensated to first order. The second-order temperature coefficient of c 11-c12 is found to decrease by 40% in magnitude when n-type doping is increased from 4.1 to 7.5 * 1019 cm -3. Results of this study enable calculation of the frequency drift of an arbitrary silicon resonator design with an accuracy of ±25 ppm between the calculated and real(ized) values over T =-40°C to +85°C at the doping levels covered in this work. Absolute frequency can be estimated with an accuracy of ±1000 ppm
Original languageEnglish
Pages (from-to)1063-1074
JournalIEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control
Volume61
Issue number7
DOIs
Publication statusPublished - 2014
MoE publication typeA1 Journal article-refereed

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Elastic constants
microelectromechanical systems
MEMS
Resonators
elastic properties
resonators
Doping (additives)
Silicon
silicon
Temperature
temperature
coefficients
Arsenic
arsenic
Carrier concentration
Boron
phosphorus
Phosphorus
boron
wafers

Cite this

@article{9332ffa032484541beb472440a9f57b2,
title = "Determination of doping and temperature-dependent elastic constants of degenerately doped silicon from MEMS resonators",
abstract = "Elastic constants c11, c12, and c44 of degenerately doped silicon are studied experimentally as a function of the doping level and temperature. First-and second-order temperature coefficients of the elastic constants are extracted from measured resonance frequencies of a set of MEMS resonators fabricated on seven different wafers doped with phosphorus (carrier concentrations 4.1, 4.7, and 7.5 x 1019 cm -3), arsenic (1.7 and 2.5 x 1019 cm-3), or boron (0.6 and 3 * 1019 cm-3). Measurements cover a temperature range from-40°C to +85°C. It is found that the linear temperature coefficient of the shear elastic parameter c11- c>sub>12 is zero at n-type doping level of n ~ 2 x 1019 cm -3, and that it increases to more than 40 ppm/K with increasing doping. This observation implies that the frequency of many types of resonance modes, including extensional bulk modes and flexural modes, can be temperature compensated to first order. The second-order temperature coefficient of c 11-c12 is found to decrease by 40{\%} in magnitude when n-type doping is increased from 4.1 to 7.5 * 1019 cm -3. Results of this study enable calculation of the frequency drift of an arbitrary silicon resonator design with an accuracy of ±25 ppm between the calculated and real(ized) values over T =-40°C to +85°C at the doping levels covered in this work. Absolute frequency can be estimated with an accuracy of ±1000 ppm",
author = "Antti Jaakkola and Mika Prunnila and Tuomas Pensala and Dekker, {James R} and Panu Pekko",
year = "2014",
doi = "10.1109/TUFFC.2014.3007",
language = "English",
volume = "61",
pages = "1063--1074",
journal = "IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control",
issn = "0885-3010",
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}

Determination of doping and temperature-dependent elastic constants of degenerately doped silicon from MEMS resonators. / Jaakkola, Antti; Prunnila, Mika; Pensala, Tuomas; Dekker, James R; Pekko, Panu.

In: IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 61, No. 7, 2014, p. 1063-1074.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Determination of doping and temperature-dependent elastic constants of degenerately doped silicon from MEMS resonators

AU - Jaakkola, Antti

AU - Prunnila, Mika

AU - Pensala, Tuomas

AU - Dekker, James R

AU - Pekko, Panu

PY - 2014

Y1 - 2014

N2 - Elastic constants c11, c12, and c44 of degenerately doped silicon are studied experimentally as a function of the doping level and temperature. First-and second-order temperature coefficients of the elastic constants are extracted from measured resonance frequencies of a set of MEMS resonators fabricated on seven different wafers doped with phosphorus (carrier concentrations 4.1, 4.7, and 7.5 x 1019 cm -3), arsenic (1.7 and 2.5 x 1019 cm-3), or boron (0.6 and 3 * 1019 cm-3). Measurements cover a temperature range from-40°C to +85°C. It is found that the linear temperature coefficient of the shear elastic parameter c11- c>sub>12 is zero at n-type doping level of n ~ 2 x 1019 cm -3, and that it increases to more than 40 ppm/K with increasing doping. This observation implies that the frequency of many types of resonance modes, including extensional bulk modes and flexural modes, can be temperature compensated to first order. The second-order temperature coefficient of c 11-c12 is found to decrease by 40% in magnitude when n-type doping is increased from 4.1 to 7.5 * 1019 cm -3. Results of this study enable calculation of the frequency drift of an arbitrary silicon resonator design with an accuracy of ±25 ppm between the calculated and real(ized) values over T =-40°C to +85°C at the doping levels covered in this work. Absolute frequency can be estimated with an accuracy of ±1000 ppm

AB - Elastic constants c11, c12, and c44 of degenerately doped silicon are studied experimentally as a function of the doping level and temperature. First-and second-order temperature coefficients of the elastic constants are extracted from measured resonance frequencies of a set of MEMS resonators fabricated on seven different wafers doped with phosphorus (carrier concentrations 4.1, 4.7, and 7.5 x 1019 cm -3), arsenic (1.7 and 2.5 x 1019 cm-3), or boron (0.6 and 3 * 1019 cm-3). Measurements cover a temperature range from-40°C to +85°C. It is found that the linear temperature coefficient of the shear elastic parameter c11- c>sub>12 is zero at n-type doping level of n ~ 2 x 1019 cm -3, and that it increases to more than 40 ppm/K with increasing doping. This observation implies that the frequency of many types of resonance modes, including extensional bulk modes and flexural modes, can be temperature compensated to first order. The second-order temperature coefficient of c 11-c12 is found to decrease by 40% in magnitude when n-type doping is increased from 4.1 to 7.5 * 1019 cm -3. Results of this study enable calculation of the frequency drift of an arbitrary silicon resonator design with an accuracy of ±25 ppm between the calculated and real(ized) values over T =-40°C to +85°C at the doping levels covered in this work. Absolute frequency can be estimated with an accuracy of ±1000 ppm

U2 - 10.1109/TUFFC.2014.3007

DO - 10.1109/TUFFC.2014.3007

M3 - Article

VL - 61

SP - 1063

EP - 1074

JO - IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control

JF - IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control

SN - 0885-3010

IS - 7

ER -