1. PART IV: EPITAXIAL SEMICONDUCTOR NANOSTRUCTURES  Properties of low-dimensional quantum confined semiconductor nanostructures  Fabrication techniques of low-dimensional semiconductor nanostructures  Formation and properties of self-assembled QDs  Growth of QWRs-QDs on patterned surfaces  Mechanisms of self ordering in epitaxial growth

2. Properties of low-dimensional quantum confined semiconductor nanostructures

3. Effect of quantum confinement on energy spectrum  Energy spectrum for electrons confined in 1, 2 or 3D with infinitely deep, rectangular potential wells with sizes t x, t y, t z :

4. Electron DOS in low-D systems Lower D  sharper DOS  potential advantage for optical and electronic properties 3D - bulk 2D - QW 1D - QWR 0D - QD

5. Sizes needed to observe QC  At T = 0K electrons occupy all energy states up to E F, corresponding to de Broglie (Fermi) wavelength F = 2  / (3  2 n) 1/3, with n = electron density.  Quantum confinement for t i ≤ F  Metals:  1 electron / atom  F ≈ 0.5nm  Semiconductors: much higher, depends on doping: e.g., n~1X10 18 cm -3  F = 29nm, t i ≈ 10nm is sufficient

6. Subband population in QC systems  If more subbands are populated, motion along confinement direction results  only ground state must be populated, i.e.,  E 12 > k B T  For infinite square QW, this means  For electrons in GaAs at T = 300K  t x < 20nm  For holes, more complicated relations and m h >m e  smaller t x  Equivalent sizes for other confinement dimensions

7. Uniformity requirements in QC structures  Size non-uniformity  inhomogeneous broadening of DOS  For ∞ wells, |  E i | / E i = 2  t i / t i ;  E i << E i   t i << t i  Practical limit to observe QC:  t i / t i < 10%   t i ≈ 1nm Calculated electron DOS in a GaAs/AlGaAs QWR with different Gaussian-shaped inhomogeneous broadening

8. Fabrication techniques of low- dimensional semiconductor nanostructures

9. From Quantum Wells to Quantum Wires/Dots ? Planar (layer-by-layer) epitaxy QWR - QD Control over lateral composition QW

10. Main approaches for creation of lateral confinement  Top-down: Post growth patterning of epitaxially grown 2D quantum wells  Bottom-up: Formation of QWR / QD during growth by special epitaxial procedures

11. Post-growth patterning 1 Advantages:  Flexibility of design (lithographic patterns) Disadvantages:  Size: several 10nm  Uniformity (size and shape): several nm  Etching defects  interface states 75-nm quantum wires fabricated in GaAs/AlGaAs material by e-beam lithography and chemical etching ( M. L. Roukes et al. Phys. Rev. Lett. 59, 3011 (1987)) SEM image showing narrow pillars etched into a GaAs substrate. (horizontal bars = 0.5  m (M. A. Reed et al., Phys. Rev. Lett. 60, 535 (1988))  Selective removal of QW by lithography, etching and regrowth  Lithography: holo, e-beam, X-ray  Etching: dry, wet depending on details of fabrication process  Regrowth: surface passivation

12. Post-growth patterning 2  Selective disordering of QWs   Patterning of QW band gap and refractive index  Methods: implantation or diffusion of impurities through a mask or with focused ion beams Advantages:  Flexibility of design (lithographic patterns) Disadvantages:  Size: several 10nm  Uniformity (size and shape): several nm  Impurities  material contamination

13. Post-growth patterning 3  Deposition of patterned “stressors” adjacent to the QW   Lateral band-gap modulation via strain effects Advantages:  Flexibility of design (lithographic patterns)  Smooth, defect-free lateral interfaces Disadvantages:  Size: several 10nm  Uniformity (size and shape): several nm

14. Post-growth patterning 4  Lateral patterning of 2D electron gas structures   Creation of QWRs, quantum point contacts (QPCs) and QDs  Methods:  Depletion by deposition of a metallic split-gate (top)  Wet chemical etching and depletion by in-plane gates (bottom) Advantages:  Flexibility of design (lithographic patterns)  Smooth, defect-free lateral interfaces  Easy electric contacts Disadvantages:  Size: several 10nm  Uniformity (size and shape): several nm

15. Cleaved-edge overgrowth  Overgrowth on the Cleaved (011) Edge of a (multiple) QW or 2DEG structure (CEO)  Cleave of the 2DEG in the MBE chamber  Overgrowth of 2DEG on top of the cleaved edge  QWR at the point where the two 2DEGs intersect   lateral variation in the potential energy  1 regrowth: QWRs; 2 regrowths: QDs After cleavage the sample is reoriented and growth is then resumed on top of the cleaved surface AlGaAs GaAs Growth direction AlGaAs GaAs The process begins with the usual growth of a high-mobility heterojunction Cleave here AlGaAs GaAs After this the sample is cleaved inside the vacuum chamber A. R. Goni et al. APL. 61, 1956 (1992)

16. Cleaved-edge overgrowth Advantages:  Size, uniformity: ML scale  Smooth, defect-free lateral interfaces Disadvantages:  Low flexibility (difficult contacts on cleaved edge) W. Wegscheider et al. PRL 71, 4071 (1993)

17. Spontaneous self-ordering 1  Growth of fractional-layer SLs on vicinal substrates  Species-dependent surface diffusion and preferential attachment of adatoms to the step edges  lateral and vertical definition, alignment  QWR formation: serpentine SL (growth rate modulations), accumulation at step bunches Advantages:  1-step process (no processing)  Size: <10nm  Lateral interfaces formed during growth Disadvantages:  Uniformity: 10-20% (imperfect step configuration and spacing, incomplete adatom segregation, growth rate variations) Stacked GaAs/AlGaAs QWR SL formed on step bunches on 3 o off (110) GaAs. (T. Kato et al., APL 72, 465 (1998)

18. Spontaneous self-ordering 2  Stranski - Krastanov growth of QDs in lattice- mismatched system (e.g., InGaAs/GaAs) Advantages:  1-step process (no processing)  Size: <10nm  Lateral interfaces formed during growth Disadvantages:  Uniformity: 10-20% in size and position (randomness of nucleation process)  Difficult contacting for transport STM image of self-assembled InAs QDs on a GaAs substrate ( M. E. Rubin et al. Phys. Rev. Lett. 77, 5268 (1996)) Improvement: growth on misoriented substrates  QD formation on quasi- periodic step edges

19. Seeded self-ordering  Growth of QWs on lithographically patterned substrates  Dielectric masks  Nonplanar surfaces  Mechanisms: selective (masks) or anisotropic (nonplanar) growth rates  material accumulation on preferential sites (“seeds”) Advantages:  Size: <10nm  Uniformity: 5% (seeds)  Lateral interfaces formed during growth Disadvantages:  2-step process (pre-patterning)  Nanostructures depend on growth habit TEM X-section of a stack of GaAs/AlGaAs QWRs grown on a V-grooved substrate GaAs V-shaped substrate 100nm AlGaAs barriers GaAs QWR

20. Formation and properties of self- assembled QDs

21. Atomic arrangement in a QD  High resolution TEM of an uncapped InAs/GaAs QD (Chu et al., JAP85, 2355 (1999))  The lateral lattice constant in the upper part of the QD is clearly larger than in the lower part: strain relaxation in the 3D island.  When too much island material is deposited, the strain cannot be totally relieved elastically through islanding, and dislocations occur via plastic relaxation.

22. Formation stages of InAs/GaAs(001) QDs 1X1  m 2 AFM scans of different InAs coverages (1 to 4 ML) on GaAs (001) (Leonard et al., PRB 50, 11687 (1994))  a) Low coverages: InAs step-flow growth.  b)-c): ~1.7ML: pseudomorphic, defect free QDs,  10% uniformity. c): Higher density, smaller size than b).  d)-f): >2ML: dislocated islands by QD aggregation or by dislocations in a single QD. Self-limiting effect

23. Critical thickness for QD formation  QD density = 0 below critical layer thickness  C  Sharp density increase after  C  QD density  =  0 (  -  C ) ,  C = 1.5ML,  = 1.76: 1st order phase transition with  an order parameter (Leonard et al., PRB 50, 11687 (1994))

24. Size distribution of QDs 1.6ML 1.65ML 1.75ML 1.9ML  Diameter and height distribution for increasing InAs coverage   10% height and  7% uniformity for initial stages of QD formation (a)  Degraded uniformity for higher   Increasing  : diameter decrease (~30nm to ~ 20nm), density increase  (Leonard et al., PRB 50, 11687 (1994))

25. Optical properties of QDs  RT PL spectra for different   2-3 peaks corresponding to ground and excited states  Size distribution of the QDs   -like DOS  broad lines (inhomogeneous broadening)  (Chu et al., JAP85, 2355 (1999))

26. Optical properties of QDs  RT PL intensity, energy and FWHM as a function of   Intensity: maximum for  ~ 2.3ML  Energy: broad minimum for  ~ 2.3- 2.7ML (  largest QDs)  FWHM: minimum for  ~ 2.6ML (30- 35meV)   larger islands: better optical quality, higher homogeneity   > 2.7ML: formation of dislocations: decreased intensity, energy shift, broader lines.  (Chu et al., JAP85, 2355 (1999)) Previous experiment: higher homogeneity, slightly higher size for lower  (first stages of QD formation)  high influence of experimental conditions!

27. Effect of growth temperature (MBE)  Increasing T (480-530C)  decreasing energy  larger QDs  Explanation: larger diffusion length  there is a larger nucleation-free area around islands (  nucleation centers, adatom sinks) where adatoms can be collected by the island  550C: In desorption (smaller QDs), In-Ga intermixing  higher energy  Increasing T: stronger, narrower lines  better material quality  Ground state – 1st subband separation (530C): ~ 70meV (Chu et al., JAP85, 2355 (1999))

28. Effect of V/III ratio (MBE)  T=480, different As 4 flux: enhanced In diffusion for lower As 4 /In ratios  Lower As 4 fluxes: increased QD quantum efficiency  Lower As 4 fluxes: small redshift  increased QD size (  larger diffusion length, coherent with T dependence) (Chu et al., JAP85, 2355 (1999))

29. Lithographic positioning of SA QDs  Self-assembled Ge islands on Si(001) pre-patterned with oxide lines  Increased uniformity in size and separation  Possible mechanisms:  Diffusion barrier on the stripe edge  Reduced strain energy at the stripe edge T. I. Kamins and R. S. Williams, APL 71, 1201 (1997)

30. Lithographic positioning of SA QDs  Preferential formation of InAs QDs in shallow, sub-  m-size GaAs holes defined by electron-beam (a) 1.4ML, b) 1.8ML InAs)  Holes with (111)A and B faces, QDs formed on B faces (favorable nucleation sites for In atoms). S Kohmoto, MSEB 88, 292 (2002)

31. Vertical stacking of QDs  Coherent InAs islands separated by GaAs spacer layers exhibit self-organized growth along the growth direction.  The island-induced evolving strain fields provide the driving force for self-assembly provided the spacer is not too thick Bright field TEM pictures taken along [011] azimuth of five sets of InAs islands separated by 36 ML GaAs spacer layers. Q. Xie et al., PRL 75, 2542 (1995) X-STM constant current topography image of two stacks of InAs QDs. D. M. Bruls et al., APL 82, 3758 (2003)

32. Lithographic positioning of stacked QDs  Twofold stacked InGaAs/GaAs QD layers grown on GaAs(001) substrates patterned with square arrays of shallow holes ((a)(-d): 100-200nm period).  The second QD layer responds to the lateral strain-field interferences generated by the buried periodic QD array: vertically-aligned QDs or satellite QDs placed on strain energy minima.  Base size and shape, and lateral orientation are predefined by the E str distribution on the underlying surface. H. Heidemeyer et al., PRL 91, 196103 (2003)

33. Growth of QWRs and QDs on patterned surfaces

34. Grating fabrication for QWRs

35. MOCVD on V-grooved substrates Stable facets forming in the groove:  sidewalls: {111}A ~ 5-10° off towards (100)  top and bottom regions: (100) + {311}A Different surface crystalline structure  different diffusion & nucleation rates  growth rate R depends on orientation R top, R bottom < R sidewall  expansion at top, sharpening at bottom BUT: profile stabilizes at the bottom at the 10nm-level {111}A (100) {311}A sidewalls GaAs substrate

36. QWR formation on V-grooved substrates AlGaAs self-limiting profile independent of lithographic details recovers after QWR deposition  ~ 10nm GaAs QW profile bottom region thickens and expands QWR formation AlGaAs vertical QW

37. Profile evolution during self-limiting growth R (100) > R {ijk}  (100) expanding R (100) > R {ijk}  (100) expanding R (100) > R {ijk}  conformal growth R (100) > R {ijk}  conformal growth layer A: t 100 > t 311 > t s  expansion of (100) and {311}A facets layer B: t 100 = t 311 = t s  stable facets, self-limiting growth layer A: t 100 > t 311 > t s  expansion of (100) and {311}A facets layer B: t 100 = t 311 = t s  stable facets, self-limiting growth G. Biasiol et al., APL 71, 1831 (1997).

38. Optical Properties of GaAs-AlGaAs QWRs * hh and lh related transitions observed polarization anisotropy in e-lh/e-hh ratio * F. Vouilloz et al. ICPS 23, Berlin, 1996 PL FWHM of QWR ~ 6meV Photoluminescence Photoluminescence Excitation

39. Mechanisms of self ordering in epitaxial growth

40. Driving force for lateral epitaxy Chemical potential ( driving force for epitaxy  supersaturation) : Lateral variations of   lateral variations of growth rate

41. Chemical potential  growth rate Diffusion towards areas of lower  Growth rate: increased at lower , decreased at higher  Nernst-Einstein relation Continuity equation

42. Example: sinusoidal chemical potential   (x)= sin (x)  j(x)  -  ’(x)= -cos(x)  R(x)   ”(x)= -sin(x) j j j j

43. How self-ordering is established Need for an equilibrating action between non-uniform chemical potential (stress, shape, composition) and another factor that drives atoms away from chemical potential minima. As growth proceeds, this should bring to steady-state growth profile. Any change in growth parameters (materials, temperature, fluxes, growth rates...) should bring to a new steady-state profile, independent of the initial one.

44. Stressed surface  self-ordering of QDs 1. SK growth mode: adatom flux towards islands  island coarsening 2. Strain energy (chemical potential) E s :  Flux away from islands  E s larger for larger islands  dissolution rate larger as island size increases 3. 1 + 2: kinetic mechanism stabilizing the island size: slowing of the growth rate of large islands and increase of the adatom density away from them, thus enhancing nucleation of new islands (with small E s  faster growth). 4.  narrow island size distribution in the system (for f = 5 and 7.5%). 1D KMC model, A.L. Barabasi, APL 70, 2565 (1997) f = 7.5% (  ) 5 (  ) 2.5% (  ) 0% (  )

45.  Pairing probability between 1st and 2nd layer of dots decreases with thicker spacers  Model: atoms of 2nd InAs layer arrive on stressed region (I) of width 2l s (  strain-driven diffusion towards top of 1st islands) or unstressed region (II) of width l-2l s (  random island formation)  l s increases as GaAs spacer is thinner   Surface diffusion model  pairing probability as a function spacer thickness, dependent on island size and density (measured), lattice mismatch and strain (calculated) and In diffusion length L D (fit parameter)  Very good match with exp data for L D = 280nm (@ T=400C) Full calculations in Q. Xie et al., PRL 75, 2542 (1995) Vertical self-ordering of stacked QDs

46. Surface chemical potential on a patterned, faceted substrate Diffusion towards the bottom Growth rate: increased at the bottom, decreased at the top µtµt µsµs µbµb Ozdemir and Zangwill, JVSTA 10, 684 (1992) lblb ltlt

47. Mechanism of self-limiting growth CapillarityGrowth rate anisotropy = Self-limiting growth G. Biasiol and E. Kapon, PRL 81, 2962 (1998), G. Biasiol et al., PRB 65, 205306 (2002).

48. Self-Limiting Growth: Al x Ga 1-x As AFM cross section of a V- groove Al x Ga 1-x As heterostructure VQW L s (Ga) > L s (Al)  stronger Ga capillarity to the bottom  Ga-rich Al x Ga 1-x As vertical quantum well L s (Ga) > L s (Al)  stronger Ga capillarity to the bottom  Ga-rich Al x Ga 1-x As vertical quantum well Nonuniform composition  ordered phase  increase of the entropy of mixing  to be included in the model Nonuniform composition  ordered phase  increase of the entropy of mixing  to be included in the model G. Biasiol and E. Kapon, PRL 81, 2962 (1998), G. Biasiol et al., PRB 65, 205306 (2002).

49. Composition dependence of self-limiting bottom width  Evidence for entropic contributions fixed by experiment fitted, L s G =175±20nm Al X Ga 1-X As; T = 700°C G. Biasiol and E. Kapon, PRL 81, 2962 (1998), G. Biasiol et al., PRB 65, 205306 (2002).

50. Temperature dependence  Arrhenius plots fit: E B G = 1.9±0.3eV fit: E B A = 2.3±0.2eV GaAs: Al X Ga 1-X As: G. Biasiol and E. Kapon, PRL 81, 2962 (1998), G. Biasiol et al., PRB 65, 205306 (2002).

51. Evolution to self-limiting profiles Modeling of experimental data; T = 700°C GaAs: Al X Ga 1-X As: G. Biasiol and E. Kapon, PRL 81, 2962 (1998), G. Biasiol et al., PRB 65, 205306 (2002).

52. QDs on etched tetrahedral pyramids QDs at the intersection of 3 QWRs  3D diffusion model [ µ(x,y) ] QDs at the intersection of 3 QWRs  3D diffusion model [ µ(x,y) ] A. Hartmann et al. APL 71, 1314 (1997) Top view