quantum transport in solids - astronomykane/pedagogical/295lec2.pdfquantum transport in solids •...
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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
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Light is a WaveHallmark of a wave : Interference
constructive
destructive
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Wave Particle Duality
A photon is a particle too
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Electrons are Waves
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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=
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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
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QuantizationWaves in a confined geometry have discrete modes
1v
2f
L=
2v2
2f
L=
3v3
2f
L=
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Quantization of circular orbits
Circular orbits have quantizedangular momentum.
2n rλ π=2 np
rπλ
= =
L rp n= =
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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
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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
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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
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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.
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Add electrons to a semiconductor
EnergyGap
Accomplished by either
• Chemical doping • Electrostatic doping (as in MOSFET)
Added electrons are mobile.
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Add electrons to a semiconductor
EnergyGap
Accomplished by either
• Chemical doping • Electrostatic doping (as in MOSFET)
Added electrons are mobile.
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Add electrons to a semiconductor
EnergyGap
Accomplished by either
• Chemical doping • Electrostatic doping (as in MOSFET)
Added electrons are mobile.
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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”
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“Holes” in the valence band
EnergyGap
Added holes are mobile
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“Holes” in the valence band
EnergyGap
Added holes are mobile
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“Holes” in the valence band
EnergyGap
Added holes are mobile
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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.
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Band Structure of a Semiconductor
p
E
Valence Band: Occupied states
Conduction Band: Empty states
electrons
holes
Gap
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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 !
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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
−
=
= ×
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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×=
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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 =