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Introduction to semiconductor nanostructures
Peter KratzerModern Concepts in Theoretical Physics: Part II
Lecture Notes
• The Fermi level (chemical potential of the electrons) falls in a gap of the band structure.
• Doping allows us to control the position of EF in the gap.
• Either electrons (n-type) or holes (p-type) act as carriers of charge.
• Long-lived optical excitations.
What is a semiconductor ?
Under which conditions does the quantum nature of the carriers show up ?
intrinsic p-type n-type
… a different answer
k
ε(k)
• σ(T) = e n(T) µ (T)• n(T) depends both on doping
and temperature
• Boltzmann statistics often sufficient to describe temp. dependence
• sometimes k ~ 0.01 alat
Basics of Transport
• conductivity σ(T) = enµ(T)• Fermi statistics,
εF~10 eV, kT << εF , kF~ alat• mobility µ: similar physics in
metals and semiconductors Drude: µ(T)=eτ(T)/m
• replace electron mass by effective mass
Is this ALL that quantum mechanics has to tell us ?
12 )(
−
∂∂∂
=ji kk
m kε
metal semiconductor
10-2 .. 105~10−2µ (cm2/Vs)
<1091021 .. 10−10>1022n (cm–3)
<10−10103 .. 10−9>104σ (Ω−1 cm−1)
insulatorsemiconductormetal
k
ε(k)
• σ(T) = e n(T) µ (T)• n(T) depends both on doping
and temperature
• Boltzmann statistics often sufficient to describe temp. dependence
• sometimes k ~ 0.01 alat―1
Basics of Transport
• conductivity σ(T) = enµ(T)• Fermi statistics,
εF~10 eV, kT << εF , kF~ alat―1
• mobility µ: similar physics in metals and semiconductors Drude: µ(T)=eτ(T)/m
• replace electron mass by effective mass
Is this ALL that quantum mechanics has to tell us ?
12 )(
−
∂∂∂
=ji kk
m kε
metal semiconductor
Excitons
• Bound system of electron and hole, cf. hydrogen atom
• Exciton radius re = a0 ε/m*1/m* = 1/me + 1/mhGaAs: re ~ 112 a0
• For structures of lateral dimensions < re, quantum confinement effects can be expected.
Nobel Prize in Physics 2000
Herbert Kroemer Zhores I. Alferov Jack S. Kilby..for developing semiconductor heterostructures ..for his part in the
in high-speed and optoelectronics integrated circuit
25 % 25 % 50 %
What is a heterostructure ?
A device build from different semiconductor materials, thus exploiting the differences in band structure.
original drawing by Herbert Kroemer, 1957
AlGaAs AlGaAsGaAs
collector base emitter
bipolar transistor
Molecular Beam Epitaxy
thermodynamics of heteroepitaxy: growth modes
• Frank-van der Merwe: ∆γ ≤ 0wetting of the substrate,layer-by-layer growth
• Volmer-Weber: ∆γ > 0no wetting, three-dimensional island growth
• Stranski-Krastanow : ∆γ ≤ 0 for the first layer(s), later ∆γ > 0 (e.g. due to lattice mismatch)island growth on the wetting layer
∆γ = γf + γi −γs
f: films: substratei: interface
Heterostructures: Band gaps/Misfits
lattice constant [Å]
Heterostructures: electrostatic potential
∆−
∆=
kTE
kTE
nekTw cc
I 2exp
2 020εε
∆EV
∆Ec EF
inversion depletion
DD Ne
kTw 202εε
=
Heterostructures: sub-bands
• Quantization of electron motion in z-direction → sub-bands
• “remote” doping → µ > 105 cm2/Vs– Ballistic motion of the electrons for d < vF τ– Fractional Quantum Hall Effect
ε2―εF > kT )(*2
)( 222
yxii kkm
++=h
εε k
From 2D to 0D: Density of States
3D
2D
1D
0D
From 2D to 1D and 0D: Practical ways
• By engineering– Lithography + etching– Cleaved-edge overgrowth– Confinement induced by
• electrostatics (gate)• STM tip, ..• strain
• By self-assembly– Colloidal quantum dots– Epitaxial quantum dots
Cleaved-edge overgrowthWidening of the potential well→ quantum wire
Colloidal CdSe Quantum dots
application: fluorescence markers in cellsnanocrystals of different sizes(different growth conditions)
wet chemical synthesis
tri-n-octyl phosphine oxide +di-methyl-cadmium
tri-n-octyl phosphine + bis-(trimethyl-silyl) selenide
1 sec
Self-Assembled Quantum Dots
Transmission electron micrograph (D. Gerthsen, TU Karlsruhe)
Epitaxial Quantum Dots: discrete DOS
cathodoluminescence temperature-independent line width
Applications
• 2D heterostructures:– high-electron-mobility transistor (HEMT) → high-
frequency electronics (cell phone, satellite TV)– solar cells with high efficiency
• Quantum dots:– light-emitting diodes, lasers – optical and IR detectors
mean free path of carriers in 2 DEG can be larger than gate length → ballistic transport
What is a laser ?
Light Amplification by stimulated emission of radiation
Requirements:• lasing medium with many objects (atoms, molecules, quantum dots, …)
capable of resonant electronic transitions• population inversion
Heterostructures in Non-Equilibriumdouble-heterostructure diode in forward bias
n-AlGaAs p-AlGaAsi-GaAs
quasi-Fermi level for electrons
quasi-Fermi level for holes
DOS ?e–
h+
strong inversion in i-GaAs !
Quantum Dot Laser
• lower threshold current than Quantum Well Laser• threshold current less temperature-dependent• varying the size and shape of the dot allows to tune emission
wavelength (without need to introduce different chemical elements)
1 ps
20-40ps
p-GaAs
p-AlGaAsp-GaAsn-GaAs
n-AlGaAs
n-GaAs
Ti-Pt-Au
Ni-Ge-Au
light-emitting layer
Semiconductor Lasers: graded-index waveguide
(110) Cleavage plane →(semi-)transparent mirrors
Semiconductor Lasers: VCSELVertical-Cavity Surface-Emitting Laser
electrical contact
upper mirror
blindlaser medium
lower mirror
electrical contactGalliumarsenide semicond. substrate
Summary
• molecular beam epitaxy → semiconductor heterostructures → band structure engineering → many novel devices
• semiconductors are an ideal playground to see quantum confinement effects, due to small electron wavevectors / large exciton radii
• self-assembled structures advantageous over “engineered” structures (small size, high density,..)
Literature
• textbooks– P. Y. Yu and M. Cardona, Fundamentals of Semiconductors,
Springer, 1996– R. Enderlin and A. Schenk, Grundlagen der Halbleiterphysik,
Akademie-Verlag, 1992 – D. Bimberg, M. Grundmann, and N.N. Ledentsov, Quantum
Dot Heterostructures, Wiley, 1999• articles
– Zh. I. Alferov, V. M. Andreev, and N. N. Ledentsov , http://link.edu.ioffe.ru/pti80en/alfer_en
– Zh. Alferov, Semiconductors 32 (1998), 1