{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T18:54:09Z","timestamp":1780685649048,"version":"3.54.1"},"reference-count":57,"publisher":"MDPI AG","issue":"21","license":[{"start":{"date-parts":[[2019,11,2]],"date-time":"2019-11-02T00:00:00Z","timestamp":1572652800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SFB 1261"],"award-info":[{"award-number":["SFB 1261"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In recent years the delta-E effect has been used for detecting low frequency and low amplitude magnetic fields. Delta-E effect sensors utilize a forced mechanical resonator that is detuned by the delta-E effect upon application of a magnetic field. Typical frequencies of operation are from several kHz to the upper MHz regime. Different models have been used to describe the delta-E effect in those devices, but the frequency dependency has mainly been neglected. With this work we present a simple description of the delta-E effect as a function of the differential magnetic susceptibility    \u03c7    of the magnetic material. We derive an analytical expression for    \u03c7    that permits describing the frequency dependency of the delta-E effect of the Young\u2019s modulus and the magnetic sensitivity. Calculations are compared with measurements on soft-magnetic        (    Fe   90     Co   10    )    78     Si   12    B  10       thin films. We show that the frequency of operation can have a strong influence on the delta-E effect and the magnetic sensitivity of delta-E effect sensors. Overall, the delta-E effect reduces with increasing frequency and results in a stiffening of the Young\u2019s modulus above the ferromagnetic resonance frequency. The details depend on the Gilbert damping. Whereas for large Gilbert damping the sensitivity continuously decreases with frequency, typical damping values result in an amplification close to the ferromagnetic resonance frequency.<\/jats:p>","DOI":"10.3390\/s19214769","type":"journal-article","created":{"date-parts":[[2019,11,4]],"date-time":"2019-11-04T04:13:08Z","timestamp":1572840788000},"page":"4769","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":46,"title":["Frequency Dependency of the Delta-E Effect and the Sensitivity of Delta-E Effect Magnetic Field Sensors"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4255-5947","authenticated-orcid":false,"given":"Benjamin","family":"Spetzler","sequence":"first","affiliation":[{"name":"Kiel University, 24118 Kiel, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Elizaveta V.","family":"Golubeva","sequence":"additional","affiliation":[{"name":"Kiel University, 24118 Kiel, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cai","family":"M\u00fcller","sequence":"additional","affiliation":[{"name":"Kiel University, 24118 Kiel, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jeffrey","family":"McCord","sequence":"additional","affiliation":[{"name":"Kiel University, 24118 Kiel, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3367-1655","authenticated-orcid":false,"given":"Franz","family":"Faupel","sequence":"additional","affiliation":[{"name":"Kiel University, 24118 Kiel, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kneller, E. 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