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