molecular beam epitaxy: principles and applications - CiteSeerX

Crystal growth of materials for energy production and energy-saving applications, R. Fornari, L. Sorba, Eds. (Edizioni ETS, Pisa, 2001)

MOLECULAR BEAM EPITAXY: PRINCIPLES AND APPLICATIONS

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G. Biasiol1 and L. Sorba1,2 Laboratorio Nazionale TASC-INFM, SS 14, Km. 163.5 Basovizza, I-34012 Trieste, Italy 2 Università di Modena e Reggio Emilia, Via Campi 213/A, I-41100 Modena, Italy e-mail: [email protected]

1. Introduction Molecular Beam Epitaxy (MBE) is an Ultra-High-Vacuum (UHV)-based technique for producing high quality epitaxial structures with monolayer (ML) control. Since its introduction in the 1970s as a tool for growing high-purity semiconductor films, MBE has evolved into one of the most widely used techniques for producing epitaxial layers of metals, insulators and superconductors as well, both at the research and the industrial production level. The principle underlying MBE growth is relatively simple: it consists essentially of atoms or clusters of atoms, which are produced by heating up a solid source. They then migrate in an UHV environment and impinge on a hot substrate surface, where they can diffuse and eventually incorporate into the growing film. Despite the conceptual simplicity, a great technological effort is required to produce systems that yield the desired quality in terms of material purity, uniformity and interface control. The choice of MBE and other growth techniques depends on the desired structure and needs. For example, in the case of mass production, MBE suffers from a lower yield, compared to other techniques such as Liquid Phase Epitaxy (LPE) and Metalorganic Vapour Phase Deposition (MOCVD), due to a lower growth rate and wafer capability (currently, GaAsbased MBE production systems are capable of up to 4X6” diameter wafers, compared to 5X10” of MOCVD). MBE, instead, is the proper technique when some particular requirements are needed, such as abruptness and control of interfaces and doping profiles, thanks to the lower growth temperature and growth rate. Besides, the control on the vacuum environment and on the quality of the source materials allows a much higher material purity, compared to non-UHV-based techniques, especially in Al-containing semiconductors for applications in high-mobility, high-speed devices (see Section 5). Finally, the UHV environment allows the use of electron diffraction probes (see Section 2.1), which provide fundamental information on the growth mechanisms. 2. Growth apparatus A schematic drawing of a generic MBE system is presented in Fig. 1. Some basic components can be identified: The vacuum system consists in a stainless-steel growth chamber, UHV-connected to a preparation chamber, where substrates are degassed prior to growth, and a load-lock module for transfer to and from air (not shown). All the components of the growth chamber must be

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able to resist bake-out temperatures of up to 200ºC for extended periods of time, which are necessary to minimize outgassing from the internal walls. The pumping system must be able to efficiently reduce residual impurities to a minimum. Typical MBE growth rates for III-V type semiconductors are of the order of 1 m/h (˜ 1ML/sec), obtained for group III partial pressures of ~10-6 Torr. With atomic densities in the crystal of about 1022 cm-3, this means that to reduce the impurity concentrations below 1015 cm-3, the impurity partial pressures must be reduced below ~10-13 Torr, assuming a unity sticking coefficient [1]. In practice, base pressure is reduced to the 10-11-10-12 Torr range, with the residual gas being essentially H2. The pumping system usually consists of ion pumps, with auxiliary Ti-sublimation and cryogenic pumps, for the pumping of specific gas species.

Fig. 1: Schematic drawing of a generic MBE system. Liquid N2 cryopanels surround internally both the main chamber wall and the source flange. Since MBE is a cold wall technique, cryopanels prevent re-evaporation from parts other than the hot cells. Besides, they provide thermal isolation among the different cells, as well as additional pumping of the residual gas. Effusion cells are the key components of an MBE system, because they must provide excellent flux stability and uniformity, and material purity. Furthermore, being the parts that must withstand the highest temperatures (up to 1400ºC) for the longest periods, they are often responsible for machine downtime. Therefore a careful choice of elements, materials and geometry must be taken. The cells (usually six to ten) are placed on a source flange, and are co-focused on the substrate heater, to optimise flux uniformity. The flux stability must be better than 1% during a work day, with day-to-day variations less than 5% [1]. This means that the temperature control must be of the order of ±1ºC at 1000ºC [2]. Furthermore, the cell geometry must be chosen in a way that the material flux does not drift appreciably as the source is depleted. The first analytical studies on flux distribution were performed on the socalled Knudsen cells, with small orifices that ensure thermodynamic equilibrium between the melt and the vapour in the cell. As a matter of fact, however, Langmuir-type (i.e., nonequilibrium) effusion cells are used in MBE growth. Due to the large orifice in these real cells, a given flux to the substrate can be reached with a lower cell temperature, resulting in a lower power consumption and in a reduction of thermal generation of impurities [1]. 67


However, in Langmuir-type cells the flux is determined not only by the vapour pressure of the material, but also by the cell geometry (see Ref. [3] for a discussion on flux calculations). A schematic drawing of a typical effusion cell is shown in Fig. 2, and some of the main features are indicated. The crucible (1) is usually made of pyrolitic boron nitride, which can stand temperatures of up to ~1300ºC without appreciable degassing. Its shape can be cylindrical or conical with different tapering angles, depending on the material to be evaporated. Its size depends on the material to be evaporated as well, and has to be big enough to provide several months of operation before the depletion of the material. Heating is provided by a Ta filament (2), while multiple Ta foils provide heat shielding (3). A thermocouple (4) is located in an appropriate position in order to measure the material temperature; temperature regulation is provided by high-precision PID regulators. A mechanical or pneumatic shutter, usually made of Ta or Mo, is placed in front of the cell to trigger the flux (see Fig. 1). The shutters must be operated much faster than the growth rate (typically 0.1 s), and should be computer-controlled to provide reproducible growth cycles, especially for superlattices. Besides, they must be designed not to outgas when heated from the cells, and not to constitute an appreciable heat shield, giving rise to flux transients after opening.

Fig. 2: Schematic drawing of a typical MBE effusion cell (Courtesy of Applied EPI, St. Paul, MN (USA)). Many variations to the basic design shown in Fig. 2 exist. An important source type for evaporation of group-V and VI elements is the so-called cracking cell. In this cell, the material is first thermally evaporated (in the form of tetramers) from a large-capacity reservoir; afterwards it passes through a hotter cracking zone in a tube, where molecules are dimerised. The effect of dimerisation on the growth mechanisms will be shown in Section 4. As far as dopants are concerned, sources, made of a thin slice of material heated by passing a DC current, are a valid alternative to effusion cells when a high material flux is not needed. This evaporation method ensures a faster response and a lower thermal load, resulting in lower contamination levels. In an effort to combine the advantages of both MBE and MOCVD, cation and/or anion solid sources in semiconductor growth have been replaced by gas sources mounted externally to the system, and have given rise to techniques such as gas source MBE (GSMBE), metalorganic MBE (MOMBE) and chemical beam epitaxy (CBE) [4]. The substrate manipulator is capable of continuous azimuthal rotation (CAR) around its axis (see Fig. 1) to improve uniformity across the wafer and from wafer to wafer in multiwafer systems. The heater behind the sample is designed to maximize temperature uniformity and minimize power consumption and impurity outgassing. Opposite to the substrate holder, an ionisation gauge is used (by rotating 180° the CAR assembly) as a beam flux monitor (BFM), for day-to-day calibration of the molecular beam intensities. On the manipulator a Mo or Ta substrate holder is mounted, onto which the wafers are either glued with an In or Ga film, or clamped with a ring. 68


Fig. 3: Picture of the high-mobility Applied EPI Gen II MBE system installed at TASC-INFM National Laboratory in Trieste, Italy [5]. Several analysis tools are available in MBE systems. The most valuable and peculiar one is Reflection High Energy Electron Diffraction (RHEED). This technique employs a high energy (up to 20 KeV) electron beam, directed on the sample surface at grazing incidence (a few degrees); the diffraction pattern is imaged on a symmetrically placed fluorescent screen. Thanks to the grazing incidence, the electron beam is scattered in the first few atomic layers, thus giving a surface-sensitive diffraction pattern. Besides, the grazing geometry avoids any interference of the RHEED apparatus with the molecular beams, making the technique suitable for analysis during growth. RHEED surface analysis will be the object of the next section. A quadrupole Residual Gas Analyser (RGA) is an essential complement to ionisation gauges, since RGA spectra provide signatures for possible air leaks, give a measure of the system cleanliness and detect impurities originating, e.g., from an insufficient system bake, or inefficient pumping, or from heating up the material sources. Additional optical, surfacesensitive analysis tools, such ellipsometry, reflectance-difference spectroscopy or laser interferometry can be optionally installed. Like RHEED, these tools are suitable for duringgrowth analysis. A discussion of their application to MBE can be found, e.g., in Ref. [3]. The growth of high electron mobility, GaAs-based heterostructures (see Section 5) requires some peculiar modifications to the system. One of such dedicated machines has been recently installed at TASC-INFM National Laboratory in Trieste, Italy (see Fig. 3 [5]). In the growth chamber, two 3000 l/s cryopumps replace the ion pumps, providing a cleaner, highercapacity pumping system. All-metal gate valves are mounted where necessary to eliminate 69


outgassing from Viton seals. No group-II materials, such as Be, are used for p-doping, since they are known contaminants that drastically reduce electron mobility. High-capacity and duplicate cells are used to avoid cell refilling or repairing for extended (up to two years) periods. In particular, a 450cc As-cracking cell has been added to two standard 125cc cells, since As2 is known to provide a more efficient incorporation onto the film and a better material quality ([6] and references therein). Extensive degassing and bake out times (3 months at 200°C) have been employed. 2.1 Reflection High Energy Electron Diffraction (RHEED) A thorough discussion of the principles of electron diffraction is beyond the scopes of this article. A more specific introduction to RHEED can be found, e.g., in Ref. [7]. A qualitative explanation of the origin of RHEED patterns can be seen in Fig. 4a). If electrons interact only with the first atomic layer of a perfectly flat and ordered surface, the three-dimensional reciprocal lattice points degenerate into parallel infinite rods. In the resulting Ewald construction the intersection of the Ewald sphere (with a radius much larger than the inter-rod spacing for typical RHEED energies) consists therefore of a series of points placed on a half circle. In reality, thermal vibrations and lattice imperfections cause the reciprocal lattice rods to have a finite thickness, while the Ewald sphere itself has some finite thickness, due to divergence and dispersion of the electron beam. Therefore, even diffraction from a perfectly flat surface results in a diffraction pattern consisting in a series of streaks with modulated intensity, rather than points (Fig. 4b)). If the surface is not flat, many electrons will be transmitted through surface asperities and scattered in different directions, resulting in a RHEED pattern constituted by many spotty features. Therefore, a first important information provided by RHEED regards the flatness of a surface. Furthermore, it is evident that diffraction from an amorphous surface (such as an oxide on top of a semiconductor) gives no diffraction pattern at all, and only a diffuse background will result. This is important, for example, for evaluating oxide desorption when a new substrate is initially heated up prior to growth in the MBE chamber, exposing the underlying, crystalline semiconductor surface.

a)

b)

Fig. 4: a) Ewald sphere construction for a two-dimensional reciprocal lattice (side view) [7]. b) RHEED geometry and formation of a diffraction pattern. A typical diffraction pattern, taken at two perpendicular azimuths, of a flat, epitaxially grown (100) GaAs surface is shown in Fig. 5. One can notice that the intensity of the streaks around the specular reflection (brightest spot) is periodic in both azimuths, with a 2x (left, [110] direction) or a 4x (right, [010]) periodicity. This modulation is caused by surface reconstruction: due to the interruption of the crystal symmetry, atoms in the topmost layers rearrange in a way to minimize their free energy, assuming an in-plane periodicity greater 70


than in the bulk. The kind of reconstruction depends critically on the material, the surface orientation and the surface termination (that in turn is a function of sample temperature and composition of the gas phase). The RHEED diffraction pattern consists therefore of main streaks with the bulk lattice periodicity (due to penetration of the beam in the first atomic layers), intercalated by weaker, surface-related lines. Therefore, RHEED can provide fundamental information about surface geometry and chemistry, both in static and dynamic (i.e., during growth) conditions.

Fig. 5: 2x (left, [110] direction) and 4x (right, [010]) diffraction patterns from a (2x4)reconstructed (001) GaAs surface. Fig. 6 plots, as a function of time, the intensity of the specular spot, in the 2x direction of a (2x4) GaAs surface, after opening the Ga (top) or the Al shutters (bottom), under an As4 flux. In both cases, the intensity initially decreases, with respect to the static conditions, and starts to oscillate periodically, with damped intensity, until shutters are closed again. The origin of the oscillations is explained in Fig. 7. If the initial surface is perfectly flat, reflectivity of the specular spot will be relatively high. As layer-by-layer growth starts, the incident electron beam gets partially scattered by the island steps of the forming monolayer, thus reducing the reflected intensity. Scattering becomes maximum at half ML coverage, while as the new monolayer completes (one Ga (Al) plus one As layer), the surface flattens again by coalescence of the islands, and the reflected intensity recovers its value. However, as shown in Fig. 6, the oscillations damp as growth proceeds, and eventually disappear. This is because at higher coverages the growth front becomes statistically distributed over more and more layers (a new ML starts before the preceding one completes), yielding eventually a constant surface roughness. The persistence of RHEED oscillations can therefore be considered as a measure of the layer-by-layer epitaxial quality. In the GaAs case (Fig. 6), after the Ga shutter is closed, the reflected intensity recovers to its initial value, due to a rearrangement of the mobile Ga adatoms as a tendency to recover surface flatness. This is not true in the AlAs case, because of the much lower surface mobility of Al atoms, as compared to Ga ones. 71


shutters open RHEED intensity (Arb. Units)

GaAs

shutters closed

AlAs

0

10

20

30

40

50

Tim e (s)

Fig. 6: RHEED intensity oscillations of the specular spot in the 2x direction during GaAs (top) and AlAs (bottom) growth.

Fig. 7: Mechanisms of RHEED intensity oscillations during growth of a monolayer [8]. Measuring the period of RHEED intensity oscillations is a powerful tool for calibrating the growth rate on a daily basis, and is one of the major advantages of MBE, in cases where an accurate precision of the layer thicknesses is needed. However, there are some precautions one should take. First, due to the above-mentioned shielding effects, there could be some growth transients when shutters are open. Second, ML steps are always present on the surface, due to unavoidable misorientations of the substrate from a low-index plane. Thus, a portion of adatoms can migrate and get incorporated at step edges. This process does not change the 72


surface roughness, and therefore part of the growth rate is not sensed by the RHEED oscillations. This step flow growth mode becomes more and more dominant over island nucleation growth as the surface is more misoriented with respect to a low-index plane, and as adatom mobility increases, i.e., at higher growth temperatures and, in GaAs, at lower As/Ga ratios. Measuring the onset for oscillation disappearance in GaAs growth, as T increases, on substrates of known misorientation, was used as a tool to infer the diffusion length and activation energy of Ga adatoms [9].

interaction potential

3. Analysis of the MBE growth process In general, three different phases can be identified in the MBE process [3]. The first is the crystalline phase constituted by the growing substrate, where short- and long-range order exists. On the other extreme, there is the disordered, gas phase of the molecular beams. In MBE, the pressures of the residual gas and of the molecular beams are so low to make the mean free path for molecule collisions much larger than the source-to substrate distance (some tens of cm). Thus MBE fluxes are ballistic and, opposite to MOCVD where fluxes are diffusive, no homogeneous reactions in the gas phase can occur. Between the two phases, a near-surface transition layer can be identified where the impinging molecular beams interact with the hot substrate (heterogeneous reactions). This is the phase where the phenomena most relevant to the MBE process take place. Atomic or molecular species get physisorbed or chemisorbed on the surface (Fig. 8a), where they can undergo different processes (Fig. 8b). Atoms can diffuse on a flat surface (a), where they can re-evaporate (b), meet other atoms to form two-dimensional clusters (c), reach a step where they can be incorporated (d), or further migrate along the step (e) to be incorporated at a kink (f). The energetics of each event, and thus their time and length scale depend on a number of factors. For example, if a surface is atomically rough, migration will be negligible. However, in normal MBE conditions, growth takes place on smooth terraces separated by ML steps. For such surfaces, the crystallographic orientation is critical and, for a given orientation, the surface reconstruction, which is determined by the temperature and the gas phase composition. As seen above, for terraced surfaces, occurrence of two-dimensional island nucleation or incorporation at steps and kinks is determined by the substrate misorientation and surface diffusion (that depends on the atomic species and fluxes, temperature, surface orientation and reconstruction).

b

physisorbed state

a)

chem isorbed state

distance from surface

a

b)

c a

e

f

d

Fig. 8: a) Interaction potential of an arriving atom or molecule with the sample surface. b) Different surface elemental processes in MBE. In heteroepitaxy (i.e., when a material is deposited over a different substrate), one has to take into account the lattice mismatch between the two materials. The strain energy due to the difference in lattice spacings is an increasing function of the growth thickness. Typically, this energy is low enough to allow a pseudomorphic, layer-by-layer growth in the first stages, where the in-plane lattice constant of the overlayer adapts to the one of the substrate. 73


However, as growth proceeds, a critical thickness is reached above which strain energy is relaxed through formation of misfit dislocations or the development of three-dimensional islands (Stransky-Krastanov growth mode). This latter phenomenon can be observed for example in SiGe/Si or InGaAs/GaAs growth with high In content and, due to the threedimensional quantum confinement properties of the resulting islands, has been used to produce self-organized quantum dots. An exhaustive treatment of lattice-mismatched growth can be found, for example, in Ref. [3]. 4. Growth of GaAs-related compound semiconductors Much of the knowledge we have today about the mechanisms of MBE is due to studies performed on GaAs growth on (001) surfaces. Some of the first fundamental studies have been performed by Foxon and Joice by means of modulated flux mass spectrometry some 25 years ago [10, 11], but their conclusions remain still valid. The recognised process of As2 incorporation (Fig. 9, left) consists in a first-order dissociative chemisorption of the physisorbed dimers on surface Ga atoms [10, 11]. By contrast, the incorporation of As4 (Fig. 9, right) involves a second-order, more complicated process: two surface tetramers must meet to generate four chemisorbed As atoms on Ga sites, and a residual As4 molecule that is desorbed [10, 11]. Therefore, the maximum sticking coefficient is 1 for As2 and 0.5 for As4. Due to the higher As species volatility, with respect to Ga, growth is usually performed with an As/Ga beam flux ratio much higher than one, or two in the case of As4 (this is a general consideration in compound semiconductor growth). This flux imbalance does not affect the one-to-one crystal stoichiometry, since As atoms do not stick if Ga atoms are not available on the surface for bonding. This means that the growth rate is ultimately determined by the Ga atoms flux, since the Ga sticking coefficient is normally one. In fact, only for extremely low As fluxes or high temperatures, outside of the so-called “MBE window” (~580-650°C), significant Ga re-evaporation takes place. Besides, the As sticking coefficient is an increasing function of the Ga flux and, with no Ga flux at all, As does not incorporate on the surface.

Fig. 9: Model of GaAs growth from Ga and As2 (left) or As4 (right) [1] Growth of IIIaIIIbV alloys, like AlGaAs, follows the same mechanisms, and an optimal growth window can be found where sticking coefficients of both group-III atoms are unity, with no mutual interference of the two species. The resulting growth rate and composition are simply derived from the two binary growth rates. Things are much more complicated in the case of IIIVaVb alloys: no unequivocal film composition can be derived from the two individual group-V fluxes, since one species is absorbed more efficiently than the other, and there is mutual interference of the sticking coefficients [12, 13]. 74


GaAs/AlGaAs systems are probably the most standard example of heteroepitaxy in MBE. Since GaAs and AlAs are lattice-matched better than 0.1%, layers of the two materials can be grown in sequence with virtually no thickness limitation for any composition. Besides, interface sharpness is normally not an issue in normal growth conditions. It is important, however, to consider and optimise interface roughness, since this is one of the factors affecting the optical quality of GaAs/AlGaAs quantum wells (QWs) and one of the scattering mechanisms affecting electron mobility in modulation-doped structures (see below). For AlGaAs-on-GaAs growth (“direct” interface), a growth interruption of a few tens of seconds yields a significant smoothing out of the interface (as can be understood by the recovery of the RHEED intensity in the top plot of Fig. 6). For the reverse (“inverted”) interface, this is true only for sufficiently low Al compositions, due to the lower mobility of Al atoms (see the bottom plot of Fig. 6). For high Al compositions, growth interruptions do not smooth appreciably the AlGaAs surface. However, in this case, the surface is rough on a length scale (a few atoms) much smaller than the exciton radius (some tens of nm), hence excitons sense an uniform average potential, yielding a relatively sharp luminescence signal [14]. Doping of semiconductors is a fundamental aspect in MBE, since it allows carrier transport in electronic or optoelectronic devices. In III-V semiconductors, doping can be achieved by using group II (p-type), IV (p- or n- type) and VI atoms (n-type). For p-type doping, elements of the II-b column (Zn, Cd) have a too high vapour pressure at usual growth temperatures, that leaves elements of the II-a column, and Be in particular, as the universal choice. GroupVI atoms are not the most common choice for n-type doping, because of surface segregation and re-evaporation problems [1]. Group-IV atoms are amphoteric, i.e., they can act as donors (if they accommodate on group-III sites) or acceptors (on group-V sites). Among these, C is an acceptor, but has a very low vapour pressure, hence it must be evaporated at very high temperatures (above 2000°C), while the amphoteric behaviour of Ge is difficult to control, and Sn presents a too high surface segregation. Therefore, the universal n-type dopant is Si, as in standard growth conditions on (Ga,In,Al)As (001) it occupies group-III sites, and doping levels up to about 1019 cm-3 can be obtained, before compensation (i.e., substitution of Si on As sites as well) is observed. One problem with Si (and Be as well) is diffusivity towards the surface at doping levels higher than about 2X1018 cm-3, which could be a problem in obtaining sharp doping profiles [15, 16]. In high mobility systems, due to the absence of group-II materials, Si is used as a p-type dopant as well, since it behaves as an acceptor by growing on GaAs(311)A surfaces in a wide range of growth conditions: it has been shown that bulk doping changes from n- to p-type by increasing the growth temperature above ~430°C, at a V-III ratios near unity [17]. 5. Magneto-transport properties of modulation doped structures The incorporation of dopants is a crucial ingredient of the device structures, but also introduces impurities in the lattice. Scattering events with these impurities cause the deterioration of the transport properties of the devices. In bulk semiconductors, the scattering mechanisms are now quite well understood and measured [18]. The scattering mechanisms can be decomposed into five contributions: 1. Optical-phonon scattering; 2. Acoustic-phonon scattering due to deformation potential; 3. Acoustic-phonon scattering due to piezoelectric field; 4. Scattering by ionised impurity; 5. Scattering by neutral impurity. The importance of the various mechanisms is shown in Fig. 10 [19] where the experimental temperature dependence of Hall mobility in n-type GaAs [20] is compared with theory [21]. It 75


is clear that at high temperatures the mobility is limited by phonon scattering whereas ionised impurity scattering dominates at lower temperatures.

Fig. 10: Temperature dependence of mobility in n-type GaAs [19]. The dashed curves are the corresponding calculated contributions from various mechanisms [21]. However, at low temperatures the limitation in the mobility can be circumvented by using the method of modulation doping proposed by Störmer et al. [22]. The idea of this method arises from the use of a structure consisting of two materials with different band-gaps grown one on top of the other. The layer sequence and the band profile for an AlGaAs/GaAs heterostructure is shown in Fig. 11. If the material with larger band-gap (barrier) is doped, in order to maintain a constant chemical potential throughout the two materials, electrons will flow from the barrier (AlGaAs) to the well (GaAs). Due to band bending at the interface, the electrons in the GaAs (well) are confined by an approximately triangular potential near the interface and form a two-dimensional electron gas (2DEG). Usually the dopants are Si atoms, which are placed in the AlGaAs layer and are separated from the interface by an undoped ‘spacer’ region. Therefore the electrons are physically separated from the ionised Si atoms, hence they are only weakly scattered by these charged impurities. modulation doping

GaAs GaAs substrate epilayer

AlGaAs

e-

GaAs cap

+ Energy

conduction band

2 DEG

EF

Fig. 11: Layer sequence and band profile of a modulation doped heterostructure. 76


In low dimensional systems, such as AlGaAs/GaAs heterojunctions or quantum wells, the same five mechanisms apply as for the bulk semiconductors. However, there are some other additional ones [23]: 6. Scattering by AlGaAs phonons; 7. Scattering by ionised or neutral impurities located in the barrier material (AlGaAs); 8. Scattering by alloy disorder; 9. Interface roughness scattering; 10. Surface phonon scattering; 11. Intersubband scattering between the quantised levels in the well. Fig. 12 shows the experimental temperature dependence of the electron mobility in a modulation doped Al0.3Ga0.7As/GaAs heterojunction [24] and the calculated contributions to the mobility from different mechanisms [25]. It is noticed that, unlike in the case of a bulk semiconductor, the mobility does not decrease as the temperature is lowered towards zero as would be expected if scattering by ionised impurities were present. A number of theoretical papers have calculated the mobility in modulation-doped structures from first principles accounting for the above-mentioned scattering mechanisms [25, 26]. Saku et al. [27] have stated that a mobility limit of about 2x107 cm2/Vs in AlGaAs/GaAs modulation doped heterostructures must exist and that the limiting scattering mechanism is due to ionised donor impurities for structures with realistic carrier concentration and optimised spacer thickness. Most models qualitatively describe the experimental results, however, they usually fail quantitatively. The weight of the different scattering mechanisms and the maximum achievable mobility are still under discussion.

Fig. 12: Mobility of modulation-doped heterostructures as a function of temperature [25]. Experimentally, a constant improvement in mobility over the years is observed in ultra-pure AlGaAs/GaAs heterojunctions, (see Fig. 13 [28]) where mobilities exceeding 107 cm2/Vs at low temperature have been achieved. The large increase is mainly associated to a constant improvement of the MBE systems and the purity of evaporation materials. Modified and extremely clean systems are employed for the MBE growth of high-mobility AlGaAs/GaAs heterostructures (see Section 2). Today, only three MBE machines in the world, dedicated to 77


the growth of high-mobility AlGaAs/GaAs heterostructures, are able to produce 2DEGs with low-temperature mobilities around 107 cm2/Vs. Up to now, the world record of 2.29x107 cm2/Vs at T 0.3T), and typical values of effective mass and scattering rate ( c SdH>10000) the oscillatory part of the resistivity can be accurately approximated as hN 2 2 m * K B T / heB ~ cos( s ) exp( ) . xx * sinh( 2 m K B T / heB ) eB c c SdH From the previous expression one can deduce the carrier density Ns from the period of SdH oscillations between two adjacent Landau levels (1/B). Ns=(e/h)/ (1/B). Furthermore, one can see from the equation that the oscillation amplitude increases with magnetic field and shrinks with increasing temperature. Fig. 17 shows SdH oscillations measured at 1.4K for a modulation-doped Al0.33Ga0.67As/GaAs heterojunction, grown at TASC-INFM [5]. From the period of the SdH oscillations, a carrier density of Ns=2.01x1011 cm-2 was obtained. The value of the carrier concentration measured by using the SdH oscillations is usually in excellent agreement with that determined by Hall measurements, provided that no parallel 80


conduction occurs in the AlGaAs barrier. Therefore, the SdH effect can be used to evaluate the parallel conductance in the AlGaAs barrier by analysing the magnitude of the drop of the longitudinal resistance. Parallel conduction is absent when the longitudinal resistance in the integer quantum Hall regime (lower magnetic fields) drops to zero. 10

Al

0.33

Ga

0.67

As/GaAs

xx

R (Ohm)

8

6

4

2

0 1

1.5

2

2.5 3 -1 1/B(Tesla )

3.5

4

Fig. 17: Shubnikov de Haas oscillations versus inverse of magnetic field for a modulation doped AlGaAs/GaAs heterostructure, grown at TASC-INFM [5]. 5.2 Quantum Hall Effect The ability to grow 2DEGs has allowed in 1980 the discovery of the quantum Hall effect (QHE) by von Klitzing at al [33]. They studied the properties of 2DEGs in a Si metal-oxidesemiconductor (MOS) structure under magnetic field. They observed experimentally that the longitudinal resistivity ( xx) vanishes for certain values of magnetic field. It is interesting to note that in correspondence of the vanishing of xx the perpendicular resistivity ( xy) exhibits plateaus at the quantised values: 1 h , xy= v e2 where the quantum number v is an integer. The quantisation of xy and the zero of xx occur whenever the Fermi level lies in between two Landau level. The existence of a finite width for the QHE plateaus and the zero longitudinal dips cannot be explained only in terms of the level quantisation. In fact if there are no states between two successive conducting Landau levels, the Fermi level should jump from the last occupied level to the next higher one and a linear increase of the resistivity with the magnetic field would be expected. Therefore, one has to invoke the existence of localised, non-current carrying, states localised in the tails of currentconducting Landau levels [19]. The broadening of the density of states in the 2DEG is well associated to the presence of disorder, nonzero temperature, and finite sample size. When varying the magnetic field the Fermi level will either lie in a delocalised, current-carrying state, with xx 0, or in a localised, not current-currying state, with xx=0 and with xy that retains a constant value due to the constant number of current-conducting carriers (see Fig. 18). Thus, the explanation of the QHE is based on a model of non-interacting particles in the presence of disorder. 81


Going to lower temperature and extremely high purity samples, novel correlation effects have been observed in the 2DEG as the Coulomb interaction between electrons exceeds their kinetic energy. New plateaus of xy and zero longitudinal dips for fractional values of v were observed (see Fig. 18). This new effect, named fractional QHE (FQHE), was discovered by Tsui el al. in 1982 [34]. The FQHE is the result of quite different physics involving strong Coulomb interaction and correlation among the electrons. The particles condense into special quantum states whose excitations have the bizarre property of being described by fractional quantum numbers including fractional charge and fractional statistics. 600

1.2 n=1.45x10

v=

3

6

5 4

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-2

6

1

2

0.8

400

(Ohm/ xx

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4/3 0.6

300 7/5

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(h/e )

5/3 8/5

xy

Resistivity

2

=2x10 cm /Vs

Resistivity

)

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11

T=340m K

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0

0 0

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6

M agnetic field B(T)

Fig. 18: Longitudinal ( xx) and perpendicular ( xy) resistivity in a high mobility, modulationdoped 2DEG grown at TASC-INFM [5], as a function of the magnetic field. Integer and fractional quantum Hall transport data are shown in Fig. 18 for a modulationdoped Al0.33Ga0.67As/GaAs heterojunction grown at TASC-INFM [5], and measured at 340mK, in collaboration with the group of F. Beltram of Scuola Normale Superiore. The numbers indicate the quantum number at which various features occur. References [1] C. T. Foxon and B. A. Joice, in Growth and Characterisation of Semiconductors, Eds. R. A. Stradling and P. C. Klipstein (IOP Publishing, 1990), p. 35. [2] K. Ploog, Angewandte Chemie 27 (1988) 593. [3] M. A. Herman and H. Sitter, Molecular Beam Epitaxy (Springer-Verlag, Berlin, 1996). [4] J. S. Foord, Chemical Beam Epitaxy and Related Techniques (John Wiley & Son Ltd, 1997). [5] http://tasc41.area.trieste.it/. 82


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