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Unformatted text preview: 6 Continued 628 6.3 Experimental methods
Crystal electron in a magnetic field Fig. 6.19 Orbit of an electron in a
solid ender the influence of an
applied magnetic field B. The orbit is
perpendicular to B and confined to
surfaces of constant energy in k
space (see eq. (6.3.1)). On closed
orbits the period is proportional to
the derivative of the crosssectional
area. A with respect to E (see eq.
(6.3.2)). In metals electrons move
on
the
Fermi
surface.
In
semiconductors they move at
surfaces of constant energy near the
minimum of the conduction band. Lorentz force: v= F = e(v x B) 1
grad k E
h &
F = hk
⇒ dk
e
= 2 B × grad k E (k )
dt h Orbit of electron:
⊥B
⊥ gradkE, i.e. on surface E = const.
629 (6.3.1) Fig. 6.20 Possible orbits of electrons in a solid in a magnetic field. For electronlike orbit the
direction of motion is as for a free electron (b). For holelike orbits the filled electron states lie outside
the Fermi surface (cf. Fig.5.23b)Therefore the direction of gradkE is such that the movement is
opossite to that of a free electron, i.e. in the direction of a positive particle (c). In addition, there are
also open orbits (a). Period on closed loop: ( 6 . 3 . 1) ⇒ T = ∫ dt
dk
e
= 2 B ( grad k E ) ⊥
dt h 1 42 43
= dE / dk ⊥ ωc = 2 T =
eB h 2π
2πeB
=2
T
h ( dA / dE ) ωc: cyclotron frequency 630 ∫ h 2
dk ⋅ dk ⊥
dA
=
dE
eB dE (6.3.2) Cyclotron frequency ωc for free electrons
h2k 2
Free electrton: E = 2m , v= k
h
m
1
grad k E = k ⇒ ⊥ =
h
m
v⊥ h Fermi surface: sphere
1 dE
= v⊥
h dk ⊥ ⇒ ωc = ⇒ e
B
m ∫ dk dk ⊥
1
=
dE
v⊥h ∫ dk = 1 2π k ⊥
2π m
=
h v⊥
h2 E.g.: B = 1T ⇒ T = 2π/ωc = 3.6 x 1011 s (6.3.3) Lorentz force
144244 = centrifuga l444
3 144 2 force
4
3
Elementary derivation:
− evB − mω 2 v ↑
=ωr Fig. 6.21 Elementary derivation of the cyclotron
frequency as a balance of Lorentz and centrifugal
force. Real space:
r: radius of the spherical or spiral
(for v ≠ 0) orbit.
Electron in solid: ωc = e
B
mc eq. (6.3.2) ⇒ h 2 dA
mc =
2π dE 631 cyclotron mass (6.3.4) Energy of electrons in magnetic field (Landau
levels)
Motion on spherical orbit: composed of two harmonic
oscillations.
⇒ discrete energy of the harmonic oscillator in plane ⊥ B.
(Landau levels of the free electron gas) hh
14243 123 2
1 E = const. + n + ωc +
k z2 .
2
2m (6.35)
harmonic
oscillator translational
in variance
in z −direction (Spin degeneration not accounted for)
Allowed states in kspace ″condence″ onto tubes (″Landau tubes″)
Fig. 6.22 Change of the density D(E)
of state of a free electron gas in a
magnetic field. According to eq. (6.3.5)
D(E) is composed of discreate energy
levels at (n + 1/ 2)hωc , i.e. δfunctions,
and the density of states of a 1dimensional electron gas perpendicular
to the direction of the applied field. In
1dimension
D(E)∝ 1 / E
(see
practical course). Since ωc = ωc(B)
D(EF) oscillates with increasing
magnetic field. ωc = ωc(B) ⇒ Oscillations of D(E) with B.
632 Fig 6.23 (a) The dots indicate the allowed states of a free
electron gas in kspace. According to eq. (6.3.5). The
electrons ″condence″ on tubes (crosssections shown as
dashed lines) in the presence of an applied magnetic field.
(b) The radius of the tubes increases with increasing
quantum number n. (c) Threedimentional sketch of the
″Landau tubes″. within the Fermi spheres. Even in
extremely strong fields the number of tubes within the
Fermi surface is much larger than shown (c) since ωc =
1.1 x 104 eV for B = 1T and EF is of the order of 5eV.
(d)The Landau tubes are only parallel to B if B is parallel
to an axis of main symetry but not for the general case
shown here. B = 1T ⇒ ω c = 1.1 × 10−4 eV ⇒
300 K : kT = 0.025eV ⇒ electrons can leave Landau tubes. 633 De Haas  van Alphen effect
Periodic oscillations of the magnetic susceptibility χmag as a
function of 1/B χmag ∝ D(EF) 1 2πe 1
∆ =
h AF B (see Chpt. 7) (see practical course) ⇒ Cross sections of Fermi surface.
Condition: kT < hω c , i.e. low temperature. 634 Cyclotron resonance in semiconductors
Fig. 6.24
Classical description of
cyclotron resonance. The electrons move
on surfaces of content energy in a static
Bfield, e.g., on the ellipsoidal surfaces,
near the conduction band minimum (see
Fig.6.4). The electrons absorbs energy
from an applied alternating high frequency
field (additional force by @Ã field) if ωHF =
ωcotherwise
acceleration
and
decceleration compensate on average. Resonance absorbtion in HF alternating field if: ω HF = ω c = e
B ⇒ mc
mc *
*
m c ⇒ m nt , m nl , m *
p Requirement: ωτ >> 1 (many periods between
collisions) (see practical course) ⇒ strong Bfield, low temperature
(conduction electrons generated
by light), pure crystals. Clear adsorbtion maximum from ″external orbits″: number of
states per frequency interval particularly high.
Resonance also for holes: opposite direction
distinguished by circularly polarized HFfield. 635 of orbits ...
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This note was uploaded on 01/19/2010 for the course MATERIALS M504 taught by Professor Adelung during the Spring '02 term at Uni Kiel.
 Spring '02
 Adelung

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