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These PowerPoint color diagrams can only be used by instructors if the 3rd Edition has been adopted for his/her course. Permission is given to individuals who have purchased a copy of the third edition with CD-ROM Electronic Materials and Devices to use these slides in seminar, symposium and conference presentations provided that the book title, author and © McGraw-Hill are displayed under each diagram. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Definition of Capacitance Qo Co V From Q = CV Co = capacitance of a parallel plate capacitor in free space Qo = charge on the plates V = voltage From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Definition of Relative Permittivity Fig 7.1 (a) Parallel plate capacitor with free space between the plates. (b) As a slab of insulating material is inserted between the plates, there is an external current flow indicating that more charge is stored on the plates. (c) The capacitance has been increased due to the insertion of a medium between the plates. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Definition of Relative Permittivity Dielectric constant Q C r Qo Co r = relative permittivity, Q = charge on the plates with a dielectric medium, Qo = charge on the plates with free space between the plates, C = capacitance with a dielectric medium, Co = capacitance of a parallel plate capacitor in free space • Insertion of dielectric induces additional charge storage thereby increase Q (and thus C) over Qo (and thus Co) • The relative permittivity (dielectric constant) is defined to reflect this increase From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Definition of Dipole Moment A dipole moment is simply a separation between a negative and positive charge of equal magnitude, Q Fig 7.2 p = Qa p = electric dipole moment, Q = charge, a = vector from the negative to the positive charge From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electronic Polarization The origin of electronic polarization. Fig 7.3 • When atom is placed in an external electric field, it will develop an induced dipole moment • The e-, which are lighter than the positive nucleus, become easily displaced by the field, which results in the separation of the negative charge center from the positive charge center • This separation induces a dipole moment and known as polarization From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Definition of Polarizability pinduced = E Induced dipole depends on the E-field causing it pinduced = induced dipole moment, = polarizability, E = electric field Electronic Polarization Z 2e2 E pe ( Ze) x β pe = magnitude of the induced electronic dipole moment, Z = number of electrons orbiting the nucleus of the atom, x = distance between the nucleus and the center of negative charge, = constant, E = electric field From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Definition of Polarizability pinduced = E Induced dipole depends on the E-field causing it Electronic Polarization • Polarization of a neutral atom involves the displacement of electrons, where is called the electronic polarization, and denoted as e • Suppose there are Z number of electrons orbiting a nucleus • When E-field is applied the light e- become displaced in the opposite direction to E, shifted from a distance x w.r.t. the nucleus at O (taken as the origin) • Although being pushed away by the E-field, coulombic attraction still exists between nucleus and electrons, pulling them back together Restoring Force Force on Electrons Z 2e2 E pe ( Ze) x β From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Static Electronic Polarizability 2 Ze e 2 me o e = electronic polarizability Z = total number of electrons around the nucleus me = mass of the electron in free space o = natural oscillation frequency From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) 1/ 2 o Zme Ze 2 e me o 2 Electronic polarizability and its resonance frequency versus the number of electrons in the atom (Z). The dashed line is the best-fit line. Fig 7.4 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) When a dilectric is placed in an electric field, bound polarization charges appear on the opposite surfaces. (b) The origin of these polarization charges is the polarization of the molecules of the medium. (c) We can represent the whole dielectric in terms of its surface polarization charges +QP and QP. • The dipoles at the edges (i.e., plates) are “uncompensated” Fig 7.5 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Definition of Polarization Vector 1 P = [p1 + p2 +... + pN ] Volume P = Polarization vector, p1, p2, ..., pN are the dipole moments induced at N molecules in the volume Definition of Polarization Vector P = Npav pav = the average dipole moment per molecule P = polarization vector, N = number of molecules per unit volume From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Polarization and Bound Surface Charge Density • Need to sum all the dipoles in the medium and divide by the volume, Ad • However, the polarized medium can be represented in terms of the surface charge +Qp and –Qp, separated by d Ptotal = Qpd • Since the polarization is defined as the total dipole moment per unit volume, the magnitude P is • However, Qp/A is the surface polarization charge density, p; therefore, P = p P = polarization, p = polarization charge density on the surface From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Definition of Electronic Susceptibility • Polarization is a vector • The direction of P is normal to the surface • For +p, comes out from the surface and for –p, goes into the surface • Although above is specific to rectangular shape, it can be generalized: • Charge/unit area on the surface of a polarized medium is equal to the component of the polarization vector normal to the surface From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Definition of Electronic Susceptibility • For Fig. 7.6, Pnormal is the component of P normal to the surface where the charge density is p, so Pnormal = p • Polarization induced in a dielectric medium when it is placed in an electric field depends on the field itself • To express the dependence of P on the field E, we define a quantity called the electric susceptibility, e by: Fig 7.6 Polarization charge density on the surface of a polarized medium is related to the normal component of the polarization vector. P = eoE Shows an effect P due to a cause E, and the quantity, e, relates the effect to its cause (proportionality constant) P = polarization, e = electric susceptibility, o = permittivity of free space, E = electric field From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electric Susceptibility and Polarization e may depend on the field itself, such that, the effect is nonlinear related to the cause. Furthermore, electronic polariziability is defined by pinduced = eE P = Npinduced = NeE Now, we can relate e and e by: e 1 o N e e = electric susceptibility, o = permittivity of free space, N = number of molecules per unit volume, e = electronic polarizability From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Relative Permittivity and Electronic Susceptibility • Important to recognize the difference between free and polarization (or bound) charges Free • The field before the dielectric was inserted is: Polarized • Where o = Qo/A is the free surface charge density • After dielectric insertion, this field remains V/d • However, free charges on the plates are different which now includes free surface charge, Q, and bound polarization charges on the dielectric surfaces next to the plates (continue on next slide) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Relative Permittivity and Electronic Susceptibility • Flow of current during the insertion of the dielectric is due to the additional free charges, Q – Qo needed on the capacitor plates to neutralize the opposite polarity polarization charges QP appearing on the dielectric surfaces • The total charge due to that on the plate plus that appearing on the dielectric surface, Q – QP, must therefore be the same as before, Qo, so that field does not change inside the dielectric: Free • Dividing by A, defining = Q/A as the free surface charge density with dielectric inserted and using eqn from previous slide: Polarized • Since P = P and P = eoE, we can elimate sP to obtain: • From definition of relative permittivity and substituting for we get: r = 1 + e r = relative permittivity, e = electric susceptibility From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Relative Permittivity and Polarizability Significance is that this relates the microscopic polarization mechanism that determines e to the macroscopic property r r 1 N e o r = relative permittivity N = number of molecules per unit volume e = electronic polarizability o = permittivity of free space Assumption: Only electronic polarization is present From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Local Field, Eloc • Previous eqn which relates r to the electronic polarizability is only approximate • Assumes field acting on an individual atom(molecule) is the field E which is assumed to be uniform within the dielectric • However, induced polarization depends on the actual field experienced by the molecule • It is evident that there are polarized Fig 7.7 molecules within the dielectric with their negative and positive charges The electric field inside a polarized dielectric at the atomic scale is not uniform. The local field is separated so that the field is not the actual field that acts on a molecules. It can be constant on the atomic scale as we calculated by removing that molecules and move through the dielectric evaluating the field at that point from the charges on the plates and the dipoles surrounding the point. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Local Field in Dielectrics and Clausius-Mossotti Eqn • The actual field experienced by a molecule in a dielectric is defined as the local field and denoted by Eloc. • It depends on the free charges on the plates and on the arrangement of all the polarized molecules around this point • To evaluate Eloc, simply remove the molecule and calculate the field at this point coming from all sources (including neighboring polarized molecules) • Therefore, crystal structure plays a role, so for a simple cubic crystal structure, the Eloc increases with polarization as: 1 E loc E P 3 o Eloc = local field, E = electric field, o = permittivity of free space, P = polarization Clausius-Mossotti Equation r 1 N e r 2 3o • The polarization can be related to Eloc and hence to E and P so P = (r – 1)oE can be used to eliminate E and P to obtain a relationship between r and e • Eqn. allows the calculation of macroscopic property r from microscopic polarization phenomena (namely, e) r = relative permittivity, N = number of molecules per unit volume, e = electronic polarizability, o = permittivity of free space From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electronic Polarization: Covalent Solids Fig 7.8 (a) Valence electrons in covalent bonds in the absence of an applied field. (b) When an electric field is applied to a covalent solid, the valence electrons in the covalent bonds are shifted very easily with respect to the positive ionic cores. The whole solid becomes polarized due to the collective shift in the negative charge distribution of the valence electrons. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electronic Mechanisms: Ionic Polarization Fig 7.9 (a) A NaCl chain in the NaCl crystal without an applied field. Average or net dipole moment per ion is zero. (b) In the presence of an applied field the ions become slightly displaced which leads to a net average dipole moment per ion. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Electronic Mechanisms: Orientional (Dipolar) Polarization Fig 7.10 (a) A HCl molecule possesses a permanent dipole moment p0. (b) In the absence of a field, thermal agitation of the molecules results in zero net average dipole moment per molecule. (c) A dipole such as HCl placed in a field experiences a torque that tries to rotate it to align p0 with the field E. (d) In the presence of an applied field, the dipoles try to rotate to align with the field against thermal agitation. There is now a net average dipole moment per molecule along the field. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Average Dipole Moment in Orientational Polarization 2 1 po E pav 3 kT pav = average dipole moment, po = permanent dipole moment, E = electric field, k = Boltzmann constant, T = temperature Dipolar Orientational Polarizability 2 1 po d 3 kT d = dipolar orientational polarizability, po = permanent dipole moment From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (a) A crystal with equal number of mobile positive ions and fixed negative ions. In the absence of a field, there is no net separation between all the positive charges and all the negative charges. (b) In the presence of an applied field, the mobile positive ions migrate toward the negative charges and positive charges in the dielectric. The dielectric therefore exhibits interfacial polarization. (c) Grain boundaries and interfaces between different materials frequently give rise to Interfacial polarization. Fig 7.11 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Total Induced Dipole Moment pav = e Eloc + i Eloc + d Eloc pav = average dipole moment, Eloc = local electric field, e = electronic polarizability, i = ionic polarizability, d = dipolar (orientational) polarizability Clausius-Mossotti Equation r 1 1 ( N e e N i i ) r 2 3 o r = dielectric constant, o = permittivity of free space, Ne = number of atoms or ions per unit volume, e = electronic polarizability, Ni = number of ion pairs per unit volume , i = ionic polarizability From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)