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Lecture 35

# Lecture 35 - EEE 352—Lecture 35 ...

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Unformatted text preview: EEE 352—Lecture 35  Magne1c Materials  Magne&c force microscope image of the bits in  IBM’s 20 Gb/in2 memory.  Philip Anderson  Sir Nevill Mo>  John van Vleck  Nobel prize in Physics, 1977, for “for their fundamental theore1cal  inves1ga1ons of the electronic structure of magne1c and disordered  systems”  Magne1c Forces and Dipoles  We have already dealt with magne1c forces in trea1ng the Hall eﬀect on  electrons and holes.  Let us review some of that work and  electromagne1cs.  Magne&c ﬂux density  Tesla = Webers/m2  Flux in Webers  Area in m2  Force created by velocity (or current)  Field created by a current:  ∫ H • dl = ∫∫ ∇ × H • ndA = ∫∫ J • ndA = I Hφ = I Ampere / meter 2πr € Magne1c Suscep1bility  Consider a cylinder of dielectric  material with a cylindrical hole (of  cross‐sec1onal area of dA) cut  down the axis of the cylinder.  The hole is ﬁlled with air, just as the  medium around the cylinder.  Magne1c Suscep1bility  In the cavity  while in the cylinder  Since the two values of the ﬂux density are equal (by construc1on),  then the ﬁeld intensi1es are related by  Hence, the necessary current (circula1ng around the hole) is  I ′ = ( H cavity − H ) dl = (µr − 1) Hdl and the magne1za1on (dipole) is  € Here, M is the magne1za1on.  (This corresponds to the “polariza1on”  in dielectric material—polariza1on arises from charge dipoles, while  magne1za1on arises from current—magne1c dipoles)  nucleus  d  Charge separa1on produces dipoles,  of value  electrons  p = qd P = Np m  Current loops produce magne1c  dipoles, of value  I  € µm = IA M = Nµm € B = µ0 ( H + M ) M = χmH µ = µ0 (1 + χ m ) = µ0µr D = ε0 ( E + P ) P = χe E ε = ε0 (1 + χ e ) = ε0εr € € There is a 1:1 duality between electric and magne1c ﬁelds (E and H),  just as between voltage and current.  That is, E and H are duals, B and D  are duals, ﬂux and charge are duals.  In the world of special rela1vity, not everything is rela&ve!  Charge and  ﬂux are not rela1ve, and do not depend upon the coordinate system, or  its mo1on.  In dielectric materials, we had several types: normal dielectric  materials, piezoelectrics, ferroelectrics, and pyroelectrics.  Similarly, there are several types of magne1c materials:   diamagne1c   paramagne1c   ferromagne1c   an1‐ferromagne1c  € χm < 1 χ m > 1, small χ m >> 1  ferrimagne1c  There are also piezomagne1c and pyromagne1c, but there use is  very small, and we will not discuss them further.  Diamagne1sm  A material is diamagne1c when   EXAMPLE: orbital electron diamagne1sm  eω 0 2 µm = IA = − (πr ) 2π orbi&ng frequency  orbit radius  Suppose we apply a small magne1c ﬁeld  € V = − ∫ Edl = E =− r dB 2 dt dΦ d , 2πrE φ = (πr 2 Bz ) dt dt € Diamagne1sm  The change in angular momentum is  Then  Larmor frequency  (equal ½ cyclotron frequency)  Now, the magne1za1on is given as  e 2r 2 M = Nµm = − B 4m = χmH e 2r 2 B = µ0 H − µ0 B 4m µ0 µ= < µ0 22 er 1+ µ0 4m € Diamagne1sm arises from a repulsive interac1on—the applied ﬁeld  slows down the orbits causing a ﬁeld which opposes the applied  ﬁeld.  This can be used to levitate diamagne1c media, because of the  opposing forces.  h>p://www.otherpower.com/diacglev.html  h>p://www.hfml.science.ru.nl/froglev.html  Paramagne1sm  A material is paramagne1c when   If the orbits are free to rotate, then they will rotate to aid the  magne1c ﬁeld.  We had previously  In an atom  so that  Bohr magneton β Paramagne1sm  Another source of magne1c moment is the spin of the electrons:  The factor gs arises because there is a strong COUPLING between  ORBITAL and SPIN angular momentum.  EXAMPLE: we calculate the spin contribu1on to the total magne1za1on.  The energy change when the ﬁeld is applied is  Each electron state is weighted by a factor determined by its gain/ loss in energy.  Electron Spin  Discovered experimentally by S. A. Goudsmit  and George Uhlenbek in 1925.  They found that the electron had an  addi1onal angular momentum, which  possessed only two values,   George Uhlenbeck  S. A. Goudsmit  Pauli realized the importance of this extra angular  momentum and postulated the exclusion principle, in  which a real space quan1zed state could have only two  electrons, with opposite spin, present.  This led to the  quantum sta1s1cs of the Fermi‐Dirac distribu1on.  Only Pauli, of this group, received the Nobel Prize!   This was awarded for the exclusion principle.  Wolfgang Pauli  Nobel prize in physics, 1945  Electron Spin  The electron spin is assumed to arise from the electron rota1ng around its own  axis.  But, which axis?  It turns out that it is whichever axis is being measured!  (But,  one can only measure one axis at a 1me.)  The two spin states can be separated by a magne1c ﬁeld, and the transi1on  between the two states is used in electron spin resonance studies of materials. The  spliing is called the Zeeman eﬀect.  In a magne1c ﬁeld (in z‐direc1on, eqn. 5.97)  Pieter Zeeman  Nobel Prize in Physics, 1902  (shared with Hendrik Lorentz  for studies in magne&sm)  g is a “fudge” factor, =2 for free electrons  This quan1ty is known as the Bohr magneton.  While it is associated with  the Zeeman eﬀect, it was discovered by Bohr and carries his name, not  Zeeman’s.  Now, let us return to the idea of the popula1on of each spin.  The energy  shim is due to the Zeeman eﬀect, as we saw earlier.  What we seek is the  spin polariza1on, or more properly, its average over all of the angles in the  solid.  Now, in a metal, most of the spins cannot turn in the magne1c ﬁeld, since  those states are occupied.  Instead, we rely upon the change in energy  1 ± gβB ≈ ± βB 2 € Parallel to the ﬁeld  Opposite to the ﬁeld  2βB 1F 1 N↑ = ∫ ρ( E + βB)dE = 2 − βB 2 1F 1 N↓ = ∫ ρ( E − βB)dE = 2 βB 2 E E F + βB € E 1 ∫ ρ( E )dE ~ 2 0 EF ∫ ρ( E )dE + 0 EF βB ρ( E F ) 2 βB ρ( E F ) 2 E F − βB 1 ∫ ρ( E )dE ~ 2 0 ∫ ρ( E )dE − 0 € Now, the magne1za1on is just  M = β ( N↑ − N↓ ) ~ β 2 Bρ( E F ) From our previous discussions of the density of states and the Fermi  energy, we have  € ρ( E ) = KE 1 / 2 F ρ ( E ) dE 2 3 N= ∫ = K ∫ ρ ( E ) dE = KE F / 2 ( E − E F ) / kB T 3 0 1+ e 0 3N 3N ρ( E F ) = ≡ 2 E F 2 kB TF ∞ E and  € 3Nβ 2 M= B 2 k B TF € Now, the magne1za1on is given as  3Nβ 2 M= B 2kB T = χmH 3Nβ 2 B = µ0 H + µ0 B 2kB T µ0 µ= > µ0 3Nβ 2 1− µ0 2kB T € New Japanese Maglev---581 km/hr The French TGV has demonstrated 574 km/hr—on track! ...
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