Ch 9. Fermi Surfaces

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Solid State Physics
Ch 9. Fermi Surfaces and Metals
Prof. J. Joo (jjoo_at_korea.ac.kr) Department of
Physics, Korea University http//smartpolymer.kore
a.ac.kr
2
9.1 Introduction

• Fermi surfaces ? the surface of constant energy
  eF in k- space ? separates the
  unfilled states from the filled states at 0K
• The electrical properties of the metal
• determined by the shape of the Fermi surface
• (? the current is due to changes in the
  occupancy of states near the Fermi
  surface)
• (?) (m)-1?d2e/dk2 or vg?de/dk
• Reduced zone scheme
• ? Always possible to select the K within the
  1st B.Z. by using a suitable reciprocal lattice
  vector G

lt Reduced Z.S gt
lt Extended Z.S gt
k
k
3
9.2 Construction of Fermi Surfaces
? In reduced zone scheme

• 1st, 2nd, 3rd B.Z.

3b
2d
2c
2a
3a
2b
2nd zone
3rd zone
? 2?? free-el.-gas? ?? Fermi surface
?
1st
? in reduced Z.S.?? 2?? free-el.-gas? ????,
3rd
2nd
1st
in periodic
free el. Fermi surface in reduced zone
4
9.3 Nearly Free Electrons

• Weakly perturbed by lattice potential
• Using the approximation constructions
• free-hand by the use of 4 facts
• Interaction with lattice ? energy gap
• Fermi surface intersects zone boundary
  (periodicity)
• Crystal potential rounds out sharp corners in the
  Fermi surfaces
• Total vol. enclosed by the Fermi surface depend
  on n and independent of the lattice
  interaction electron concentration

2nd
3rd
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Solid State Physics
Ch 10. Plasmons, Polaritons, and Polarons
Prof. J. Joo (jjoo_at_korea.ac.kr) Department of
Physics, Korea University http//smartpolymer.kore
a.ac.kr
6
10.1 Plasma Optics

• ? Dielectric Function of Electron Gas
• Displacement (CGS unit)
• ? Eq. of motion of free electron in electric
  field (no bound and no damping)
• ? Plasma a medium with equal concentration of
  positive and negative charges of which one
  charge type is mobile.
• In solid, the negative charges of conduction
  els. Are balanced by an equal concentration of
  positive charges of ion cores.

? Positive ion background dielectric const.
?(8) ?gtgt?p?
??
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10.2 Dispersion Relation for Electromagnetic
Waves

• In non-magnetic isotropic medium, the EM eq.
• ??? ??
• If e real and gt0. For ? real, k is real. And
  transverse EM wave propagates with the phase
  velocity
• If e real and lt0. For ? real, k is imaginary.
  The wave is damped with a characteristic length
• If e complex. For ? real, k is complex. The
  wave is damped.
• e 8 . Finite response without applied field.
• e 0 longitudinal polarized wave ? later

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10.3 Electrostatic Screening

• Consider a positive charge embeded in el. gas
• el. gas tends together around and thus to screen
  the positive charge
• static screening described by e(0, k )
• Thomas-Fermi dielectric function
• Screened Coulomb Potential

r
screened the potential!
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10.4 Mott Metal - Insulator Transition

• Hydrogen atom ? half-filled ? metal
• Hydrogen molecule ? filled ? insulator
• At T0K, H-atom is insulator or metal ?
• depends on a
• ac 4.5 a0 Bohr radius
• considering
• At high concentration, ks? large, then
• e-ksr factor is weak! ? possible to be
  metal!
• ???, ks is small ? insulator!
• Polaritons
• ? Quantum of the coupled phonon-photon
  transverse wave.
• due to transverse optical phonons and
  transverse EM waves
• ? Note longitudinal phonons do not coupled
  to transverse photons in crystal

Fig.10b
metal
insulator
s (S/cm)
2
4
6
n, in 1018 cm-3

1. M-I transition depends on the extrinsic
   parameters such as pressure, composition,
   magnetic field, temp.
2. Insulating phase suggest el.-el.
   interaction (Coulomb gap.)

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Solid State Physics
Ch 11. Optical Process and Excitons
Prof. J. Joo (jjoo_at_korea.ac.kr) Department of
Physics, Korea University http//smartpolymer.kore
a.ac.kr
11
11.1 Introduction (1)

• Rapid progress of use of laser for band structure
  ?optical spectroscopy
• from Far-IR, IR (infrared), visible,
  ultraviolet
• the most important intrinsic property of matter
• e(?)er(?)iei(?) ??
• Not directly accessible experimentally
  from optical measurements.
• (???) obtained from reflectance (R) or
  transmittance
• refractive
  index n(?)
• extinction
  coefficient K(?)
• ? er(?)n2-K2
• ei(?)2nK
• reflectivity coefficient r(?)

UV/Vis
difficult to find!
phase
amplitude
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11.1 Introduction (2)
Reference A. M. Appendix K (p.776)

• ?
• ???, ????? ?? ??? ? reflectance R(?)
• ?(?) can be calculated from the measured
  reflectance R(?), if R(?) is
  known at all frequencies.
• ltKramers-Kronig Relationgt
• 
  p principal part of integral
• 
  s parameter

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11.1 Introduction (3)
11.2 Hagen-Rubens Formula or Approximation

• Summary to find e(?)er(?)iei(e)
• Measure R(?) ? ? 0 ? 8 , if possible
• Use k-k relation
• Known R(?), ?(?)
• Use r(?)R1/2(?)e i?(?)
• Find n(?) and k(?)
• ? er(?)n2-k2 and ei(?)2nk

• Rrr
• for (transparent) dielectric
• Kltlt1 ? K ?0
• for metal, K is large and n is also large
• nK
• (in the limit of low freq.)

Reference Introduction to modern physics by
Fowles p. 168
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11.3 Excitons (1)

• lt Electronic Interband Transition gt
• Direct interband absorption of photon h?
• lt Excitons gt
• ?? bound el.-hole pair (due to attractive
  Coulomb interaction)
• Exciton can move through the crystal and
  transport energy. But it does NOT transport
  charge. (?electrically neutral)
• All excitons are unstable w.r.t. the ultimate
  recombination process, in which the el. drops
  into the hole in the valence band, accompanied by
  the emission of photon or phonon.

Eg
e-
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11.3 Excitons (2)

• In the formation of excitons, the energy is lower
  w.r.t. the threshold (Eg) by the binding energy
  of the excition (1meV1eV)
• 3 ways to measure the exciton binding energy
• Optical transitions from the V.B. ?required
  energy Eg-Ex
• Recombination luminescence
• Photo-ionization of excitons ? to form free
  carriers
• (?? 7 ??)
• lt Exciton condensation into el.-hole drops (EHD)
  gt
• Exciton decay time 8µs
• However, exciton condensation at low temp.
• ? condensed into a drop ? EHD life-time 40µs

C.B.
Eg
Eex exciton binding energy
exciton energy level
h?
V.B.
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Solid State Physics
Ch 13. Dielectrics and Ferroelectrics
Prof. J. Joo (jjoo_at_korea.ac.kr) Department of
Physics, Korea University http//smartpolymer.kore
a.ac.kr
17
13.1 Questions

• What is the relation between the dielectric
  polarization p and the macroscopic el. field E?
• What is the relation between the dielectric
  polarization p and the local el. field which acts
  at the site of an atom in the lattice?
• ? polarization P dipole moment per unit vol.
• To find the contribution of the polarization to
  macroscopic field, simplify the sum over all
  dipoles in the sample.
• Total macroscopic el. field
• applied due to uniform
  polarization
• field (?depolarization field)
• E.g.) Fig.4 tends to oppose the applied
  field (E0)

E0
E0
P
E1
E0
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13.2 Dielectric Constant and Polarizability (1)

• Dielectric constant e of isotropic or cubic
  medium w.r.t. vacuum is defined as
• or
• Susceptibility ?
• Polarizability (a) of an atom is defined
• in terms of the local field P aEloc
• dipole ???? atom ??? local field
• moment polarizability
  (atomic property)
• Polairzation
• polarization dipole moment
• Relation between e and a
• microscopic ? local el.
  field
• macroscopic property

magnetic
or
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13.2 Dielectric Constant and Polarizability (2)

• relation of dielectric constant to a (related
  to crystal structure)

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13.3 Structural Phase Transition

• Some of crystals usually transform from one
  crystal structure to another as the T (??), P
  (??), or external fields.
• Exhibits an el. dipole moment even in the absence
  of an el. field
• ? hysteresis
• Piezoelectricity PZd?E
• (?? ??)

13.4 Ferroelectric Crystals
P?E
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Reference..
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Solid State Physics
Ch 14. Dielectrics and Ferroelectrics
Prof. J. Joo (jjoo_at_korea.ac.kr) Department of
Physics, Korea University http//smartpolymer.kore
a.ac.kr
23
14.1 Introduction

• Money for magnet gt Money for semiconductor
• Magnetism Q.M.S.M.
• ?? 1 ?? ??
• 3 principal sources of the mag. Moment of a free
  atom
• Electrons (spins)
• El. orbit angular momentum about the nucleus
• Change of the orbital moment induced by an
  applied mag. Field
• Magnetization M mag. moment per unit vol.
• ? (??) P polarization
• Mag. susceptibility
• Molar susceptibility (?M) ? ?/mass or ?/mol
• Negative ? diamag.
• Positive ? paramag.
• Ordered arrays of mag. moments ferro - ,
  ferri - , antiferro -

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14.2 Langevin Diamagnetism Eq. (1)

• Diamagnetism ? origin the magnetic field of the
  induced current
• is opposite to the applied field
• Mag. moment of el.
• ???? ????, Lorentz force ? ??

neucleus
Ze
e-
F0
??? ??

1. Rotation of el. has been slowed down
2. Reduction of freq. corresponding charge in mag.
   field

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14.2 Langevin Diamagnetism Eq. (2)

• Consider Ze nucleus (Z electrons)
• Mag. moment

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14.3 Paramagnetism

• Electronic paramagnetism positive ?
• Atoms, molecules, and lattice defects possessing
  and odd number of electrons total spin ? 0
• Free atoms and ions with a partially filled inner
  shell transition elements, rare earth elements
• Metals (except some of diamagnetic - )

27
14.4 Quantum Theory of Paramagnetism

• Mag. moment
• Energy levels in a mag. field
• p.427 ltCrystal field splittinggt ?? 6 ??

spin
orbital