Introduction to semiconductor nanostructures Peter Kratzer Modern Concepts in Theoretical Physics: Part II Lecture Notes
What is a semiconductor? The Fermi level (chemical potential of the electrons) falls in a gap of the band structure. Doping allows us to control the position of E F in the gap. intrinsic p-type n-type Either electrons (n-type) or holes (p-type) act as carriers of charge. Long-lived optical excitations. Under which conditions does the quantum nature of the carriers show up?
a different answer
metal Basics of Transport semiconductor conductivity σ(t) = enµ(t) σ(t) = e n(t) µ (T) Fermi statistics, n(t) depends both on doping ε and temperature F ~10 ev, kt metal << ε F, k F ~ a lat semiconductor insulator ε(k) mobility µ: similar physics in metals and semiconductors σ (Ω 1 Drude: cm 1 ) µ(t)=eτ(t)/m >10 4 10 3.. 10 9 <10 10 replace electron mass by k effective mass 1 2 n (cm 3 ) >10 ( ) m = 22 ε k 10 21.. 10 10 <10 9 Boltzmann statistics often k i k j sufficient to describe temp. dependence µ (cmis 2 /Vs) this ALL that ~10 quantum 2 10-2.. 10 sometimes 5 k ~ 0.01 a lat mechanics has to tell us?
Basics of Transport metal conductivity σ(t) = enµ(t) Fermi statistics, ε F ~10 ev, kt << ε F, k F ~ a 1 lat mobility µ: similar physics in metals and semiconductors Drude: µ(t)=eτ(t)/m replace electron mass by effective mass 1 2 ( ) ε k m = k i k j Is this ALL that quantum mechanics has to tell us? semiconductor σ(t) = e n(t) µ (T) n(t) depends both on doping and temperature ε(k) k Boltzmann statistics often sufficient to describe temp. dependence sometimes k ~ 0.01 a lat 1
Excitons Bound system of electron and hole, cf. hydrogen atom Exciton radius r e = a 0 ε/m* 1/m* = 1/m e + 1/m h GaAs: r e ~ 112 a 0 For structures of lateral dimensions < r e, quantum confinement effects can be expected.
Nobel Prize in Physics 2000 25 % 25 % 50 % Herbert Kroemer Zhores I. Alferov Jack S. Kilby..for developing semiconductor heterostructures..for his part in the in high-speed and optoelectronics integrated circuit
What is a heterostructure? A device build from different semiconductor materials, thus exploiting the differences in band structure. bipolar transistor AlGaAs GaAs AlGaAs collector base emitter original drawing by Herbert Kroemer, 1957
Molecular Beam Epitaxy
thermodynamics of heteroepitaxy: growth modes γ = γ f + γ i γ s Frank-van der Merwe: γ 0 wetting of the substrate, layer-by-layer growth f: film s: substrate i: interface Volmer-Weber: γ > 0 no 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
Heterostructures: Band gaps/misfits lattice constant [Å]
Heterostructures: electrostatic potential E c E F E V inversion depletion w I = εε 0 kt 2 e n 2 0 E kt c exp Ec 2kT wd = 2 2εε 0 kt e N D
Heterostructures: sub-bands Quantization of electron motion in z-direction sub-bands ε 2 ε F > kt ε i ( k ) = ε i + h 2 m 2 * ( k x 2 + k y 2 ) remote doping µ > 10 5 cm 2 /Vs Ballistic motion of the electrons for d < v F τ Fractional Quantum Hall Effect
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 overgrowth Widening of the potential well quantum wire
Colloidal CdSe Quantum dots wet chemical synthesis tri-n-octyl phosphine + bis-(trimethyl-silyl) selenide 1 sec tri-n-octyl phosphine oxide + di-methyl-cadmium application: fluorescence markers in cells nanocrystals of different sizes (different growth conditions)
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) highfrequency 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-Equilibrium double-heterostructure diode in forward bias e DOS? quasi-fermi level for electrons h + quasi-fermi level for holes n-algaas i-gaas p-algaas strong inversion in i-gaas!
1 ps Quantum Dot Laser 20-40ps 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)
Semiconductor Lasers: graded-index waveguide p-gaas p-algaas p-gaas n-gaas n-algaas n-gaas Ti-Pt-Au light-emitting layer (110) Cleavage plane (semi-)transparent mirrors Ni-Ge-Au
Semiconductor Lasers: VCSEL Vertical-Cavity Surface-Emitting Laser electrical contact upper mirror blind laser medium Galliumarsenide semicond. substrate lower mirror electrical contact
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