16. Dielectrics / Metals

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1 Technische Universität Graz Institute of Solid State Physics 16. Dielectrics / Metals Nov. 26, 2018

2 Dielectrics Dielectrics used as electrical insulators should not conduct. Large breakdown field. Low AC losses. Sometimes a low dielectric constant is desired (CMOS interconnects) Sometimes a high dielectric constant is desired (supercapacitors).

3 Breakdown field Typically V/cm

4 AC losses - loss tangent In an ideal capacitor, current leads voltage by 90 o. Because the dielectric constant is complex, in real materials current leads voltage by 90 o -. Power loss = 2 1V0 tan 2 Becomes more of an issue at high frequencies (microwaves) 0 0

5 Loss tangent

6 Polarizability Overdamped modes Orientation polarizability Space charge polarizability Underdamped modes Ionic polarizability Electronic polarizability

7 Orientation (dipolar) Polarizability For materials (gases, liquids, solids) with a permanent dipole moment. The theory is very similar to paramagnetism. 1 T Curie law

8 Orientation Polarizability Ion jumps. doubly ionized

9 Water Schematic dielectric function of water from Wikipedia Source: Classical Electrodynamics, J.D. Jackson

10 Ionic Polarizability Displacement of ions of opposite sign. Only in ionic substances. This is an underdamped mode in the infrared.

11 Electronic polarizability (all materials) P Np N E dipole moments density polarizability

12 Polarizability Kittel

13 Inter- and intraband transitions When the bands are parallel, there is a peak in the absorption (") E ( k) E ( k) c v Optical spectroscopy has developed into the most important experimental tool for band structure determination. - Kittel

14 Ashcroft and Mermin

15 Dielectric function of BaTiO3

16 Dielectric function of silicon absorption ( ) n( ) ik( ) refractive index n 4 fk c extinction K

17 AC Conductivity For constant voltage, conductors conduct and insulators don't. For low ac voltages in a conductor, electric field and the electron velocity are inphase, electric field and electron position are out-of-phase. For low ac voltages in an insulator, electric field and the electron position are inphase, electric field and electron velocity are out-of-phase. At high (optical) frequencies the in-phase and out-of-phase component of the response is described by the dielectric function.

18 Conductivity / Dielectric function Harmonic dependence,, v v e x x e E E e i t i t i t v i x 0 0 P nex E E j nev inex E E E i i 0 Below about 100 GHz the frequency dependent conductivity is normally used. Above about 100 GHz the dielectric function is used (optical experiments).

19 Diffusive transport (low frequencies) v F ma ee m d e sc E v m E v e d d sc sc m e ne 2 ne m

20 Diffusive metal The current is related to the electric field j E n nm m v E n nm m Steady state solution The differential equation that describes how the velocity changes in time is: Inertial term m dv() t ev() t dt ee() t The impulse response function : 1 et g ( t) exp t 0 m m

21 Diffusive metal The differential equation is: m dv() t ev() t dt ee() t Assume a harmonic solution E()e it, v()e it i m 1 v( ) E( ) e v E 1 i m 1 1 1i 1 i 2 2 e 1 1 ne j ne v i E E m e Scattering time low ne high ine m 2

22 Diffusive metal ne 1 i 2 2 i0 i0 1 ne i i 1p p 2 ne m 0 Take the limit as goes to infinity 2 p for 0 for 0

23 low frequency metal / high frequency insulator Ibach & Lueth Aluminum ITO Conducting transparent contacts for LEDs and Solar cells Windows that reflect infrared 2 p 2 ne m 0 Reflection of radio waves from ionosphere

26 Ellipsometry n sin n sin

27 Ellipsometry Ellipsometry measures the change of polarization upon reflection. The measured signal depends on the thickness and the dielectric constant.

28 Ellipsometry The ratio of the two reflected polarizations is insensitive to instabilities of light source or atmospheric absorption.

Lecture contents Review: Few concepts from physics Electric field

1 Lecture contents Review: Few concepts from physics Electric field Coulomb law, Gauss law, Poisson equation, dipole, capacitor Conductors and isolators 1 Electric current Dielectric constant Overview