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 EF 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 Basics of Transport metal semiconductor • conductivity s(T) = enm(T) • s(T) = e n(T) m (T) • Fermi statistics, • n(T) depends both on doping e ~10 eV, kTmetal << e , k ~ a semiconductorand temperatureinsulator F F F lat e(k) • mobility m: similar physics in metals and semiconductors -1 -1 4 3 -9 -10 s (W Drude:cm )m(T)=et(T)/m>10 10 .. 10 <10 • replace electron mass by k effective mass -1 n (cm–3) >1022æ ¶2e (k) ö 1021 .. 10-10 <109 m = ç ÷ ç ¶k ¶k ÷ • Boltzmann statistics often è i j ø sufficient to describe temp. dependence m (cm2/Vs) ~10-2 10-2 .. 105 Is this ALL that quantum • sometimes k ~ 0.01 a mechanics has to tell us ? lat Basics of Transport metal semiconductor • conductivity s(T) = enm(T) • s(T) = e n(T) m (T) • Fermi statistics, • n(T) depends both on doping e ~10 eV, kT << e , k ~ a ―1 and temperature F F F lat e(k) • mobility m: similar physics in metals and semiconductors Drude: m(T)=et(T)/m • replace electron mass by effective k mass -1 æ ¶2e (k) ö m = ç ÷ ç ¶k ¶k ÷ • Boltzmann statistics often è i j ø sufficient to describe temp. dependence Is this ALL that quantum • sometimes k ~ 0.01 a ―1 mechanics has to tell us ? lat Excitons • Bound system of electron and hole, cf. hydrogen atom • Exciton radius re = a0 e/m* 1/m* = 1/me + 1/mh GaAs: re ~ 112 a0 • For structures of lateral dimensions < re, 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 f: film Dg = g + g -g s: substrate f i s i: interface • Frank-van der Merwe: Dg £ 0 wetting of the substrate, layer-by-layer growth • Volmer-Weber: Dg > 0 no wetting, three-dimensional island growth • Stranski-Krastanow : Dg £ 0 for the first layer(s), later Dg > 0 (e.g. due to lattice mismatch) island growth on the wetting layer Heterostructures: Band gaps/Misfits lattice constant [Å] Heterostructures: electrostatic potential DEc EF DEV inversion depletion ee kT DE DE 2ee 0kT 0 c æ c ö w = wI = 2 expç- ÷ D 2 2e n0 kT è 2kT ø e N D Heterostructures: sub-bands • Quantization of electron motion in z-direction → sub-bands 2 e (k ) = e + h ( k 2 + k 2 ) e2―eF > kT i i 2 m * x y • “remote” doping → m > 105 cm2/Vs – Ballistic motion of the electrons for d < vF t – 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 • By self-assembly – Lithography + etching – Colloidal quantum dots – Cleaved-edge overgrowth – Epitaxial quantum dots – Confinement induced by • electrostatics (gate) • STM tip, .. • strain 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 nanocrystals of different sizes application: fluorescence markers in cells (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) → 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-Equilibrium double-heterostructure diode in forward bias e– DOS ? quasi-Fermi level for electrons quasi-Fermi level h+ for holes strong inversion n-AlGaAs i-GaAs p-AlGaAs in i-GaAs ! Quantum Dot Laser 1 ps 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 Ti-Pt-Au light-emitting p-GaAs layer p-AlGaAs p-GaAs n-GaAs n-AlGaAs (110) Cleavage plane → n-GaAs (semi-)transparent mirrors Ni-Ge-Au Semiconductor Lasers: VCSEL Vertical-Cavity Surface-Emitting Laser electrical contact upper mirror blind laser medium lower mirror Galliumarsenide semicond. substrate 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.