1.6 Magnetic Couplings
When piezomagnetic effects are present and need to be considered, the piezoelectric
constitutive relations in Eq. (1.2) 1,2 can be generalized to [12]
T ij ¼ c ijkl S kl À e kij E k À h kij H k ,
D i ¼ e ikl S kl þ ε ik E k þ α ik H k ,
B i ¼ h ikl S kl þ α ik E k þ μ ik H k ,
ð1:46Þ
where B is the magnetic flux or induction, H the magnetic field, h ijk the
piezomagnetic constants, α ij the magnetoelectric constants, and μ ij the magnetic
permeability. B satisfies ∇ Á B ¼ 0. Under the well-known quasistatic approximation,
∇ Â H ¼ 0. Then a magnetic potential ψ can be introduced through H ¼ À ∇ ψ.
With the presence of the electric field, magnetic field, temperature gradient, electric
conduction, and heat conduction, there are many cross effects such the thermomagnetic, Hall, and Ettingshausen effects [12]. Most of them are beyond the scope of the
present book.
References
1. H.F. Tiersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969)
2. A.H. Meitzler, D. Berlincourt, F.S. Welsh III, H.F. Tiersten, G.A. Coquin, A.W. Warner, IEEE
Standard on Piezoelectricity (IEEE, New York, 1988)
3. B.A. Auld, Acoustic Fields and Waves in Solids, vol 1 (Wiley, New York, 1973)
4. J.S. Yang, An Introduction to the Theory of Piezoelectricity, 2nd edn. (Springer, New York,
2018)
5. R.F. Pierret, Semiconductor Device Fundamentals (Pearson, Uttar Pradesh, 1996)
6. S.M. Sze, Physics of Semiconductor Devices (Wiley, New York, 1981)
7. H.G. de Lorenzi, H.F. Tiersten, On the interaction of the electromagnetic field with heat
conducting deformable semiconductors. J. Math. Phys. 16, 938–957 (1975)
8. G.A. Maugin, N. Daher, Phenomenological theory of elastic semiconductors. Int. J. Eng. Sci.
24, 703–731 (1986)
9. S. Selberherr, Analysis and Simulation of Semiconductor Devices (Springer, New York, 1984)
10. R.D. Mindlin, On the equations of motion of piezoelectric crystals, in Problems of Continuum
Mechanics, ed. by J.R.M. Radok, (Society for Industrial and Applied Mathematics, Philadelphia, 1961), pp. 282–290
11. R.D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates.
Int. J. Solids Struct. 10, 625–637 (1974)
12. A.C. Eringen, G.A. Maugin, Electrodynamics of Continua, vol I (Springer, New York, 1990)
13. D.M. Rowe, CRC Handbook of Thermoelectrics (CRC Press, Boca Raton, 2016)
References
11
When piezomagnetic effects are present and need to be considered, the piezoelectric
constitutive relations in Eq. (1.2) 1,2 can be generalized to [12]
T ij ¼ c ijkl S kl À e kij E k À h kij H k ,
D i ¼ e ikl S kl þ ε ik E k þ α ik H k ,
B i ¼ h ikl S kl þ α ik E k þ μ ik H k ,
ð1:46Þ
where B is the magnetic flux or induction, H the magnetic field, h ijk the
piezomagnetic constants, α ij the magnetoelectric constants, and μ ij the magnetic
permeability. B satisfies ∇ Á B ¼ 0. Under the well-known quasistatic approximation,
∇ Â H ¼ 0. Then a magnetic potential ψ can be introduced through H ¼ À ∇ ψ.
With the presence of the electric field, magnetic field, temperature gradient, electric
conduction, and heat conduction, there are many cross effects such the thermomagnetic, Hall, and Ettingshausen effects [12]. Most of them are beyond the scope of the
present book.
References
1. H.F. Tiersten, Linear Piezoelectric Plate Vibrations (Plenum, New York, 1969)
2. A.H. Meitzler, D. Berlincourt, F.S. Welsh III, H.F. Tiersten, G.A. Coquin, A.W. Warner, IEEE
Standard on Piezoelectricity (IEEE, New York, 1988)
3. B.A. Auld, Acoustic Fields and Waves in Solids, vol 1 (Wiley, New York, 1973)
4. J.S. Yang, An Introduction to the Theory of Piezoelectricity, 2nd edn. (Springer, New York,
2018)
5. R.F. Pierret, Semiconductor Device Fundamentals (Pearson, Uttar Pradesh, 1996)
6. S.M. Sze, Physics of Semiconductor Devices (Wiley, New York, 1981)
7. H.G. de Lorenzi, H.F. Tiersten, On the interaction of the electromagnetic field with heat
conducting deformable semiconductors. J. Math. Phys. 16, 938–957 (1975)
8. G.A. Maugin, N. Daher, Phenomenological theory of elastic semiconductors. Int. J. Eng. Sci.
24, 703–731 (1986)
9. S. Selberherr, Analysis and Simulation of Semiconductor Devices (Springer, New York, 1984)
10. R.D. Mindlin, On the equations of motion of piezoelectric crystals, in Problems of Continuum
Mechanics, ed. by J.R.M. Radok, (Society for Industrial and Applied Mathematics, Philadelphia, 1961), pp. 282–290
11. R.D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates.
Int. J. Solids Struct. 10, 625–637 (1974)
12. A.C. Eringen, G.A. Maugin, Electrodynamics of Continua, vol I (Springer, New York, 1990)
13. D.M. Rowe, CRC Handbook of Thermoelectrics (CRC Press, Boca Raton, 2016)
References
11