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

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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

    Fingerprint

    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",
    publisher = "IEEE Institute of Electrical and Electronic Engineers",
    number = "7",

    }

    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 -