User:MaterialsPsych/Aluminium gallium arsenide

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Structural Properties
AlGaAs possesses a zincblende crystal structure, like its parent compounds GaAs and AlAs, although it can be distorted slightly into a tetragonal structure when cooled down from the high temperatures it is grown at.

Like other compound zincblende semiconductors, aluminium and gallium occupy half of the sites in the crystal structure (the yellow atoms in the diagram), while other half (grey atoms) are occupied by arsenic. The aluminium and gallium atoms are usually randomly distributed across the sites they occupy through the crystal, although it is possible to obtain long-range spatial ordering of the aluminium and gallium atoms (i.e., a superstructure) by manipulating growth conditions.

The cubic lattice parameter of AlGaAs depends on its composition and temperature. At 300 K, the following relationship between the lattice constant $$a$$ (in angstroms) and composition of AlxGa1-xAs ($$0 < x < 1$$) has been recommended: $$a(x) = 5.6533 + 0.00809x$$ Similarly, the density of AlGaAs is a function of its composition and temperature. At 300 K, the following relationship between the density $$d$$ (in g cm-3) and composition of AlxGa1-xAs $(0 < x < 1)$ has been recommended: $$d(x) = 5.3165 - 1.5875x$$

At approximately 900 C, AlGaAs has nearly the same lattice constant as the GaAs that is grown on. However, once cooled to 300 K, the cubic lattice constants of AlGaAs, GaAs (5.6533 Å) and AlAs (5.6611 Å) differ from GaAs (5.6533 Å) by less than 0.2%. Although this mismatch is still small, it can induce substantial bowing in GaAs wafers with very thick aluminium-rich AlGaAs layers grown on them, which can be deleterious when later processing the wafers into devices. To compensate for this, a small amount of phosphorus is sometimes included in the growth to substitute for arsenic atoms, forming arsenic-rich aluminium gallium arsenide phosphide, which reduces, but does not completely eliminate, wafer bowing.

Electronic properties
Like GaAs and AlAs, AlGaAs is a semiconductor with properties generally falling between those of its parent compounds.

Band gap
As a semiconductor, AlGaAs has positive, nonzero band gap. The band structure and band gaps of AlxGa1-xAs fall between those of pure GaAs and AlAs, varying with composition and temperature. The room temperature (300 K) direct and indirect band gaps of GaAs and AlAs are tabulated below, indicating the extrema of the band gaps in AlGaAs at x = 0 and x = 1. The nearby figure depicts a schematic the basic band structure of GaAs and AlAs, showing the direct (Γ-Γ) and indirect (Γ-X and Γ-L) band gaps in the first Brillouin zone.

The following relations have been suggested for the direct and indirect band gaps in AlxGa1-xAs as a function of composition at 300 K: $$E_{\Gamma-\Gamma} = 1.310x^{3} - 1.437x^{2} + 1.708x + 1.423$$ $$E_{\Gamma-X} = 0.055x^{2} + 0.210x + 1.899$$ $$E_{\Gamma-L} = 0.645x + 1.707$$

Since pure GaAs is a direct band gap semiconductor, while AlAs is an indirect band gap semiconductor, the band gap in AlxGa1-xAs transitions between direct and indirect at an intermediate composition. This transition occurs at approximately x = 0.4.

Carrier mobility
The carrier mobility in AlGaAs and its heterostructures is an important parameter for electronic device design and performance. Broadly speaking, the electron mobility characterizes how fast electrons flow through a material when an electric field is applied, while the hole mobility characterizes how fast holes flow through a material under an applied electric field. Carrier mobilities are frequently inferred via measurements of the Hall effect, yielding the Hall mobility, which is used here.

In AlxGa1-xAs films with n-type dopants, electron mobilities have generally been observed to be higher with a smaller Al fraction in the alloy. For x < 0.3, electron mobilities ranging from approximately 3000 to 5000 $E_{g}(0) = 1.519 eV$ at 300 K have been reported, with the mobilities generally increasing up to a maximum value dependent on the dopant level as the temperature is decreased. With increasing Al fraction in the alloy, the electron mobility generally decreases. The reported electron mobilities of Al-rich samples with x > 0.4 sharply drops to less than 1000 $α = 5.405 × 10^{-4} eV/K$.

The hole mobilities of doped AlxGa1-xAs are much lower than the electron mobilities. The maximum hole mobilities in AlGaAs at 300 K are around 250 $β = 204 K$. Similar to the electron mobility, the hole mobility is observed to gradually decrease with increasing $cm^{2}/(V⋅s)$, falling to less than 100 $cm^{2}/(V⋅s)$ for AlGaAs samples with large fractions of aluminium.

It is important to distinguish the carrier mobility in modulation-doped AlGaAs/GaAs heterostructures from the carrier mobility in AlGaAs itself. As is explained in further detail below, the electrons in a two-dimensional electron gas formed in a modulation-doped AlGaAs/GaAs heterojunction are localized in the GaAs layer near the AlGaAs/GaAs interface. Therefore, the carrier mobilities in these heterostructures are derived primarily from carrier transport in the undoped GaAs layer, not the doped AlGaAs layer. Far higher carrier mobilities are obtained in AlGaAs/GaAs two-dimensional electron and hole gases compared to those in AlGaAs itself. Electron and hole mobilities up to $cm^{2}/(V⋅s)$ and $x$, respectively, have been measured in AlGaAs/GaAs electron and hole gases at cryogenic temperatures.

Heterojunction band offsets
When a heterojunction between GaAs and AlGaAs (or AlAs) is formed, there exists an energy offset between the conduction and valence bands of the two layers. The band offsets of GaAs/AlGaAs heterojunctions have been studied in detail. GaAs and AlGaAs form a "type I" heterojunction as shown in the figure for any composition of AlGaAs, where the valence band maximum of GaAs is above that of AlGaAs and the conduction band minimum of GaAs is below that of AlGaAs. The magnitude of the discontinuities in the conduction and valence bands at the heterojunction are denoted by $cm^{2}/(V⋅s)$ and $3.5 × 10^{7} cm^{2}/(V⋅s)$, respectively. The following relationships have been recommended for $5.8 × 10^{6} cm^{2}/(V⋅s)$ and $ΔE_{c}$ in electronvolts as a function of AlxGa1-xAs composition: $$\Delta E_{V} = 0.51x \; (0 \leq x \leq 1)$$ $$\Delta E_{C} = 0.80x \; (0 \leq x < 0.45)$$

Another source recommends the following relations: $$\Delta E_{V} = 0.46x \; (0 \leq x \leq 1)$$ $$\Delta E_{C} = \begin{cases} 0.79x & (0 \leq x < 0.41) \\ 0.475 - 0.335x + 0.143x^{2} & (0.41 < x \leq 1) \end{cases} $$

Dopants and defects
Doping is a technologically important way of modifying the electronic properties of an intrinsic semiconductor, accomplished by introducing impurity atoms into the semiconductor during growth to make it an extrinsic semiconductor. N-type dopants "donate" additional electrons into the conduction band, so they are called donors. P-type dopants "accept" electrons from the valence band to form holes, so they are called acceptors. Donors and acceptors form additional, discrete energy levels within the band gap.

Many of the dopants used in GaAs can be utilized similarly in AlGaAs. For example, some of the chalcogens, such as sulfur, selenium and tellurium, are n-type dopants in AlGaAs. Beryllium, magnesium and zinc are p-type dopants.

In theory, Group IV elements, such as carbon, silicon, germanium and tin can act as either donors or acceptors in III-V semiconductors, depending on which of the two sites in the crystal lattice they occupy, making them "amphoteric" dopants. If they occupy a group V site (i.e., arsenic), they will act as acceptors. Conversely, if they occupy a group III site (i.e., aluminium or gallium), they will act as donors. A mixture of both leads to the donors and acceptors compensating for each other. In practice, carbon preferentially acts as an acceptor in GaAs and AlGaAs. Silicon is an strongly amphoteric dopant in GaAs and AlGaAs; whether it predominantly acts as a donor or acceptor has a complex dependence on the density of silicon atoms incorporated into the GaAs/AlGaAs, growth direction, temperature and other growth conditions. The maximum level of useful silicon doping in GaAs and AlGaAs as a donor is limited to about $ΔE_{v}$ as a result of silicon's amphoteric behavior and the presence of DX centers (discussed more in the next paragraph).

In GaAs and Ga-rich AlGaAs, many of the donors discussed above form energy levels near the conduction band minimum, making them shallow donors. However, when x > 0.22, lowest energy level for all single atom donors (e.g., Si, S, Se, Te) becomes a donor level called the DX center, which lies at an energy level further down in the band gap, thus making all single atom donors deep donors in Al-rich AlGaAs. The microscopic character and origin of DX centers have been the subject of extensive study due to the (usually negative) impact that DX centers have on device performance, and many different hypotheses have been put forth as to their origin and microscopic structure. Experimental signatures of DX centers have also been observed in AlGaAs with x < 0.22 or GaAs under certain conditions, such as in samples that are heavily doped or subjected to elevated pressures. DX centers limit the maximum concentration of free carriers introduced by donors in GaAs and AlGaAs to approximately $ΔE_{c}$ at standard atmospheric pressure.

Point defects in the crystal structure of GaAs and AlGaAs, such as vacancy, interstitial and antisite defects can also effectively act as donors or acceptors. For example, if an As atom substitutes for a Ga or Al atom, thereby creating a AsAl/Ga antisite defect, it will act as a deep donor, creating an energy level approximately 0.97 eV above the valence band in AlGaAs that is independent of the AlGaAs composition.

Optical properties
Because AlGaAs is used in optoelectronics, such as LEDs and laser diodes, its optical properties have been extensively studied. The complex refractive index of AlGaAs, which can be used to directly calculate many other quantities, is primarily a function of the AlGaAs composition, the wavelength of the incident light and temperature. Adachi has compiled tables of the complex refractive indices of AlxGa1-xAs as a function of photon energy and its composition $ΔE_{v}$ at a temperature of 300 K, which are plotted here as functions of wavelength and composition. The real $2 × 10^{19} cm^{-3}$ and imaginary $2 × 10^{19} cm^{-3}$ parts of the complex refractive index $(x)$ are plotted here separately: $$\tilde{n} = n + i\kappa$$ The linear attenuation coefficient $(n)$ can be directly obtained from the imaginary part of the refractive index at any wavelength $(κ)$ according to the equation: $$\alpha = \frac{4 \pi \kappa}{\lambda}$$

Preparation
Most published literature regarding AlGaAs preparation focuses on growing films of single crystal AlGaAs on gallium arsenide substrates. An overview of the most prevalent growth techniques is provided below.

Metalorganic vapor phase epitaxy
AlGaAs thin films have also been prepared by atomic layer epitaxy, which is a variation of MOVPE that allows for the precise growth of very thin layers by pulsing the precursors to the substrate one at a time.

Liquid phase epitaxy
AlGaAs-GaAs heterostructures have also been prepared by liquid phase epitaxy (LPE). Many different variations of the LPE method and its application to AlGaAs growth have been developed and described over time. Mauk asserts that LPE was the preferred technique for growing AlGaAs/GaAs heterostructures for LEDs and laser diodes in the 1970s and 1980s, but goes on to state that it has since fallen out of favor compared to MOVPE (for larger scale industrial production) and MBE (for smaller scale research). Nonetheless, LPE is still reported to be used by some companies and in some academic research labs.

The LPE growth of AlGaAs on GaAs substrates typically first involves preparing a mixture of gallium, aluminium and arsenic at elevated temperatures in a graphite crucible, also called a "boat". Reported growth temperatures range from approximately 350°C to 800°C, with lower temperatures (350°C - 550°C) typically being utilized to grow thinner layers (<10 nm) than could be successfully grown at higher temperatures, which are used to grow thicker (>100 nm) AlGaAs layers. Molten gallium acts as a solvent to dissolve aluminium and arsenic. Solid aluminium readily dissolves into molten gallium, while arsenic is usually supplied from a sacrificial piece of GaAs in the melt. Once the precursors have been mixed and heated to form a uniform, saturated solution, a GaAs substrate is lowered into the boat and placed in contact with the melt. In the "slow-cooling" method, which is generally the most common method that is employed, the crucible is slowly cooled down at a constant rate, which causes the solution to become slightly supersaturated, leading to the precipitation and growth of AlGaAs on the GaAs substrate. Several other schemes for ramping the temperature down, using a temperature gradient or using other external driving forces to induce and control the liquid-phase epitaxial growth of AlGaAs on GaAs have also been described.

Due to the tendency for oxygen to react with the precursors at growth temperatures and for oxide impurities to interfere with crystal growth, the LPE growth of AlGaAs is typically carried out under a hydrogen atmosphere that is continuously purged. Published procedures usually call for the gallium and GaAs to be annealed under a hydrogen atmosphere at temperatures slightly higher than those used for growth for extended periods of time prior to growth to bake out residual oxygen and oxide impurities. It is also suggested that aluminium be treated with hydrogen fluoride, for example, to remove the native layer of oxide from its surface before it is added to the melt.

Like with MBE and MOVPE, LPE also allows for the controlled doping of AlGaAs layers to produce n- and p-type doped layers. Germanium, magnesium, and, less commonly, zinc, may be added as precursors to be incorporated into the AlGaAs as p-type dopants, while tin and tellurium are employed as n-type dopants.

It is possible to grow several different AlGaAs layers sequentially with different dopants and aluminium concentrations by using an apparatus with multiple graphite boats that each contain a different mixture of precursors. The substrate is moved sequentially between boats to grow the different layers. This style of apparatus is described as being "the industry standard for the LPE growth of most compound semiconductor structures" circa 1989.

Condensed matter physics research
GaAs/AlGaAs heterostructures have played an important role in fundamental condensed matter physics research since the early 1980s. The fractional quantum Hall effect (FQHE) was discovered in 1982 by Daniel Tsui and Horst Störmer in a GaAs/AlGaAs heterostructure that was grown by Arthur Gossard using molecular beam epitaxy. A schematic of the heterostructure that is described in their original paper is shown in the nearby figure. This is very similar to the situation described above with the high-electron-mobility transistor, as a two-dimensional electron gas is formed in the GaAs near the GaAs/AlGaAs interface, with the electrons coming from the silicon atoms in the doped AlGaAs layer. When Tsui and Störmer applied a large magnetic field to a device (i.e., a Hall bar) on the sample at extremely low temperatures, less than 5 K, they unexpectedly observed a dip in the transverse resistance ($(ñ)$) that coincided with a newly-discovered plateau in the Hall resistance, $α$, in addition to the other plateaus in the Hall resistance that came from the then-known integer quantum Hall effect. The existence of this plateau in the Hall resistance could not be explained by the theory that had been previously developed to explain the integer quantum Hall effect. Coupled with the subsequent development of a theory by Robert Laughlin in 1983 that explained their observation of this Hall resistance plateau, Störmer, Tsui and Laughlin were jointly awarded the Nobel Prize in Physics in 1998 for their work.

Since then, GaAs/AlGaAs heterostructures have continued to be an important tool for studying the physics of the FQHE. Some of the highest electron mobilities recorded in any material system have been obtained in MBE-grown GaAs/AlGaAs heterostructures in the pursuit of studying the FQHE, with reported electron mobilities exceeding $λ$, which can be compared to the value of $R_{xx}$ reported with Tsui, Störmer and Gossard's original FQHE discovery. Similarly, record-breaking hole mobilities approaching $R_{xy} = 3h⁄e = 3R_{K} ≈ 77.4 kΩ$ have been reported in GaAs/AlGaAs heterostructures.