View
218
Download
0
Category
Preview:
Citation preview
Quantum Transport in Solids
• Waves and particles in quantum mech.
• Quantization in atoms
• Insulators : Energy gap
• Emergent particles in a solid
• Landau quantization in a magnetic fieldand the quantum Hall effect
Light is a WaveHallmark of a wave : Interference
constructive
destructive
Wave Particle Duality
A photon is a particle too
Electrons are Waves
Energy and Momentum
Max Planck 1858-1947
Louis de Broglie 1892-1987
Planck : Energy ~ Frequency
E hf ω= =
hp kλ
= =
de Broglie: Momentum ~ (Wavelength)-1
34/ 2 1.05 10h Jsπ −= = ×
2 fω π=
2 /k π λ=
Dispersion RelationRelation between
• Frequency and Wavelength• Energy and Momentum
For a photon:
cf ck E cpωλ
= ⇒ = ⇒ =
2 2 221 v
2 2 2p kE mm m
ω= = ⇒ =
For an electron in vacuum:
p
E
p
E
QuantizationWaves in a confined geometry have discrete modes
1v
2f
L=
2v2
2f
L=
3v3
2f
L=
Quantization of circular orbits
Circular orbits have quantizedangular momentum.
2n rλ π=2 np
rπλ
= =
L rp n= =
Bohr Model
Niels Bohr 1885-1962
vL m r n= =2 2
2
ve mF kr r
= =
2/nE Ry n= −2
0nr n a=
2 4
2 13.6 eV2
mk eRy = =
2
0 2 .5amke
= = Å
• Two equations :
• Bohr radius :
• Rydberg energy:
• Solve for rn and En(rn,vn)
Explains line spectra of atoms
Flatland ... (1980’s)Semiconductor Heterostructures : “Top down technology”→ Two dimensional electron gas (2DEG)
Ga As
AlxGa1-x As
2DEGz
V(z)
∆E ~ 20 mV ~ 250°K
2D subband
conduction bandenergy
Fabricated with atomic precision using MBE. 1980’s - 2000’s : advances in ultra high mobility samples
z
Pauli Exclusion Principle
Wolfgang Pauli 1900-1958
Enrico Fermi 1901-1954
Electrons are “Fermions”:
Each quantum state canaccommodate at most one electron
n=1
n=2
n=3
n=4
n=1
n=2
n=3
n=4
Ground state Excited State
Insulator
EnergyGap
An insulator is inert because a finite energy is required to unbind an electron.
A semiconductor is an insulator with a smallenergy gap.
Add electrons to a semiconductor
EnergyGap
Accomplished by either
• Chemical doping • Electrostatic doping (as in MOSFET)
Added electrons are mobile.
Add electrons to a semiconductor
EnergyGap
Accomplished by either
• Chemical doping • Electrostatic doping (as in MOSFET)
Added electrons are mobile.
Add electrons to a semiconductor
EnergyGap
Accomplished by either
• Chemical doping • Electrostatic doping (as in MOSFET)
Added electrons are mobile.
Added electrons form a band
p
E2
2 *pEm
=
m* = “Effective mass”
Electrons in the “conduction band” behave just likeordinary electrons with charge e, but with a renormalized mass.
“Emergent quasiparticles at low energy”
“Holes” in the valence band
EnergyGap
Added holes are mobile
“Holes” in the valence band
EnergyGap
Added holes are mobile
“Holes” in the valence band
EnergyGap
Added holes are mobile
Holes are particles too
p
E2
2 *h
pEm
=
mh* = Hole Effective mass
Holes in the “valence band” behave just likeordinary particles with charge + e, with a mass mh* .
The sign of the charge of the carriers can be measured with the Hall effect.
Band Structure of a Semiconductor
p
E
Valence Band: Occupied states
Conduction Band: Empty states
electrons
holes
Gap
The Quantized Hall Effect
Von Klitzing,1981 (Nobel 1985)
Hall effect in 2DEG MOSFET at large magnetic field
• Quantization:
• Quantum Resistance:
• Explained by quantum mechanics of electrons in a magnetic field
/xy QR Nρ =2/ 25.812 807 kQR h e= = Ω
N = integer accurate to 10-9 !
Quantization in a magnetic fieldCyclotron Motion:1D Quantization argument:
vL m r n= =2vv mF e B
r= =
ceBm
ω =
nnreB
=2 2n ceB nE nm
ω= =
Cyclotron frequency
Magnetic flux enclosed by orbit
Solve for rn and En(rn,vn)
BF
v
e
vF e B= − ×
202n
n nr Beππ φ= = Magnetic Flux
quantum 0
15 24.1 10
he
Tm
φ
−
=
= ×
Landau Levels
Lev Landau 1908-19681( )2n cE nω= +
Landau solved the 2D SchrodingerEquation for free particles in a magnetic field
Closely related to the harmonic oscillator
/ cE ω
Each Landau level has one stateper flux quantum
total flux Area# states = flux quantum ( / )
Bh e×=
Quantized Hall Effect
yxy
x
E BJ ne
ρ = =
/ cE ω N filled Landau levels:# particles/area:
Gap
0
# flux quantaarea
n N
B eBN Nhφ
= ×
= =
From last time:
2
1xy
hN e
ρ =
Recommended