Lecture 3: Heterostructures, Quasielectric Fields, and Quantum Structures MSE 6001, Semiconductor Materials Lectures Fall 2006 3 Semiconductor Heterostructures A semiconductor crystal made out of more than one material, where composition varies with position but the crystal periodicity of atoms is kept, is referred to as a heterostructure. 3.1 Semiconductor alloys Single-crystal semiconductor alloys can be made that have bandgaps and other properties in between those of the end point materials. Figure 1 illustrates how the bandgap of Al y Ga 1 y As, with 0 y 1, has a bandgap energy E g in between those of AlAs and GaAs. Grading the composition, that is varying the concentration with position, gives spatially-varying semiconductor properties. The Al y Ga 1 y As alloy system The AlAs/GaAs family of materials has been extensively developed for its use in heterostructures. These materials have the same crystal structure, the cubic zinc blende crystal lattice, and almost exactly the same lattice constant a (the periodic repeat length in the crystal). AlAs and GaAs are also completely miscible, and all values of y in Al y Ga 1 y As are stable. 3.2 Quasi electric fields. Consider a GaAs wafer that has a surface layer of Al y Ga 1 y As that is graded from y = 0.3 at the surface where x = 0 to y = 0 at x = x o. This composition profile and the resulting spatially GaAs Al y Ga 1-y As AlAs E C E C E C E V E V E V FIGURE 1: Bandgap energies for the materials GaAs, Al y Ga 1 y As, and AlAs. 3-1
y 0.3 0 x 0 x E E C = 0.263 ev E C (x) E V (x) FIGURE 2: Linearly graded material layer. varying bandgap are given in Fig. 2. The slope in the conduction band edge E C (x) corresponds to a quasi electric field, E. Recall that for a real electric field E, E = 1 q de C dx = 1 q de V dx. (1) The units of the electric field are [V/cm]. In the absence of an applied or internal electric field, the conduction band quasi electric field due to composition changes is defined as, E C = 1 q For linear grading of composition, as in Fig. 2, de C dx. (2) de C dx = E C x. (3) Note that for the AlAs/GaAs system, the conduction band and valence band quasi electric fields have opposite signs. In our example above, E C = 0.263 ev for y = 0.3. If x is taken as 100 nm, then the quasi electric field is EC = 2.63 10 4 V/cm. Conduction band electrons would experience this field and be accelerated in the positive x-direction. Note that the quasi electric field for the valence band will in general have a different magnitude and can have a different sign. In this case, the valence band quasi electric field has a value EV = +1.32 10 4 V/cm. [For material parameters, see Tiwari and Frank, APL 60 630 (1992).] 3-2
%i- w- 2 -E r aa Es 1.s?' a$ &E s $ o w r3!g I GaP I I 1 I I T=300K Ee.- CL-' Ezn %S 5, AIP I I I I -^ ^^ ^. s.ti t5.u b.4 Lattice Constant (A) FIG. 1. The conduction band edge and valence band edge energies plotted as a function of the lattice constant of semiconductors. The circles indicate the band edges of the binary semiconductors and the lines show the band edges of the ternary alloys. The two endpoints of each ternary line are the binary constituents of that ternary. Discontinuities between two lattice matched or nearly matched semiconductor alloys may be found from the difference in energy between their band-edge energies. The zero energy point represents the approximate gold Schottky barrier position in the band gap of any given alloy. FIGURE 3: Band edges versus lattice constant for III-V semiconductors. [Tiwari and Frank, APL 60 630 (1992).] 3.3 Materials Families Only small lattice constant mismatches may be accommodated by strain in a heterostructure. Figure 3 displays band edges versus lattice constant for most of the III-V semiconductors. A family of materials refers to the alloy compositions that may be grown on a given substrate, such as InAs. Lattice constants generally need to match within about 1%, depending on layer thicknesses. 3.4 Quantum heterostructures Heterostructure layers may be thin enough that the conduction band electrons and valence band holes are quantum-mechanically confined to a layer with lower bandgap. This confinement changes the electron and hole energies, giving a bandgap that depends on the well width: a quantum well. (Fig. 4) A periodically repeating sequence of quantum wells and barriers is called a superlattice. For a deep quantum well, where the energy of the confined state is well below the top of the confining barrier, the confinement energy of the n th energy level in the conduction band well, E Cn may be estimated with the expression for an infinitely-deep 1D quantum well of width l, E n = n2 h 2 π 2 2m n l 2. (4) This energy of the lowest-confined state energy, E C1, defines the slightly raised, lower edge of the conduction band in the quantum well material. The electron mass m n is the conduction band effective mass of the quantum well material. The confinement in valence band, E V n, depends 3-3
E E C (x) E C1 l ψ(x) x x FIGURE 4: Semiconductor quantum well. on the effective mass of the valence band holes, m h, and the highest-confined state energy, E V 1, defines the slightly lowered, upper edge of the valence band in the quantum well material. For GaAs, the effective masses are m e = 0.067m e and m h = 0.082m e. The new bandgap of the quantum well material becomes E g(qw ) = E g + E V 1 + E C1. (5) Quantum wells are used, for example, to engineer the lasing energy (light color) of semiconductor lasers. They are also used to engineer the electronic transport, for example, by doping with donors in the barrier of the well, from which the electrons fall into the well to give a 2D sheet of charge. The 2D electron gas (2DEG) can move in the sheet with very high conductivity because there are no dopant atoms in the sheet to interrupt the motion. Such manipulation of the optical and electrical properties with heterostructures is refered to as bandgap engineering. 3.5 Semiconductor nanostructures Quantum well confinement effects become important for layers that are on-the-order of 10 nm or thinner in most materials. For most of the column-iv and III-V semiconductors, the cubic lattice constants lay between 0.45 nm and 0.65 nm, which corresponds to a thickness of two molecular layers, giving quantum well thicknesses of some 40 molecular layers or less. Figure 5 gives a cross sectional image formed with scanning-tunneling microscopy (STM) of the atoms in a super lattice made with InAs and GaSb layers. If instead of a thin sheet of material, a thin wire of semiconductor is formed, then it is referred to as a quantum wire or a semiconductor nano wire. Figure 6 gives a SEM micrograph of a silicon nanowire grown at the University of Utah by Sun-Gon Jun and a diagram of a nanowire transistor under development by Justin Jackson. If only a small volume with 10-nm-scale dimensions 3-4
is formed, then is is referred to as a quantum dot. Figure 7 displays an atomic force microscope (AFM) image of strained InAs quantum-dot islands grown in etched GaAs trenches and a transmission electron microscope (TEM) cross-sectional image through strained GaAs nanowires on a GaP substrate. InSb like InSb like [001] GaSb GaSb GaSb GaSb InAs InAs InAs M. Weimer Texas A&M University FIGURE 5: STM image of the cleaved edge of an InAs/GaSb superlattice. 3-5
Nanowire MOSFET n-poly Si drain wrap around poly Si gate nanowire channel spin on glass, PECVD SiO 2 n-si source p-si substrate ~ 100 nm (a) (b) FIGURE 6: An electron micrograph of silicon nanowires (a) and a schematic cross section through a nanowire transistor (b). The smallest lithographic features needed are 100 nm for the catalyst and gate etch. (S. G. Jun and M. S. Miller, U Utah). 1 µm dot chain (a) (b) FIGURE 7: Strained quantum dots and quantum wires. (a) An electron micrograph of InAs quantum dot islands in GaAs trenches, and (b) TEM of GaAs wires on GaP substrate. (B. Ohlsson, Lund U., and M. S. Miller, U Utah). 3-6