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4.12 Modification of Bandstructure: Alloys and Heterostructures. Since essentially all the electronic and optical properties of semiconductor devices are dependent on the bandstructure, how can we change the bandstructure of a material to achieve desirable electronic and optical properties?
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4.12 Modification of Bandstructure: Alloys and Heterostructures Since essentially all the electronic and optical properties of semiconductor devices are dependent on the bandstructure, how can we change the bandstructure of a material to achieve desirable electronic and optical properties? In principle, many physical phenomena can modify the bandstructures, but there are two important ones that are widely used for band tailoring. One involves alloying of two or more semiconductors and another is the use of heterostructures to form “superlattices”.
Alloys of Semiconductors • When two semiconductor A and B are mixed via an appropriate growth technique, one has the following properties of the alloy: • The crystalline structure of the lattice: In most semiconductors the two (or more) components of the alloy have the same crystal structure so that the final alloy also has the same crystalline structure. For the same lattice structure materials the lattice constant obeys the Vegard’s law for the alloy AxB1-x, • where x is the average fraction of A atoms and 1-x is the average fraction of B atoms in the alloy, aA and aB are the lattice constants of A and B. • Bandstructures of alloys: The bandstructure of alloys is difficult to calculate in principle since alloys are not perfect crystals even if they have a perfect lattice. This is because the atoms are placed randomly and not in any periodic manner. A simple approach used for the problem is called the virtual crystal approximation. According to this approximation, the bandstructure of alloy AxB1-x is simply the weighted average of the bandstructures of A and B and given by
where EA and EB are the bandstructures of A and B, respectively. Effective masses: From equation 12.2, we can express the effective electron (hole) mass of an alloy by Above relations are strictly valid if there is a good “mixing” in the alloy formation process. Thus if an alloy AxB1-x is grown, the probability that an A-type atom is surrounded by a B-type atom should be (1-x) and the probability that B-type atom is surrounded by a A-type atom should be x. When this is true, the alloy is called a random alloy. Otherwise, A-atoms or B-atoms form clustered or become phase separated.
Deviations from the random alloy model: bowing. • According to equation 12.2, band gap of an alloy should a linear function of x. Deviations from the simple linear relation usually occur and called bowing. As a result the linear equation is modified as • where EB is the bowing parameter, a quantity determined experimentally. One example is given in the figure below for GaAs and InAs alloy whose bowing parameter is 0.37. The variation of the energy-band gap as a function of the indium mole fraction in InGaAs. The dotted line is the result of “virtual-crystal approximation”m while the solid line is the experimental curve and clearly “bows” away from the linear relation.
Semiconductor heterostructures • In addition to alloying, fabricating artificial structures with tailored optical and electronic properties has been possible using crystal growth techniques, such as molecular beam epitaxy (MBE) and metal-organic chemical vapor deposition (MOCVD). These techniques allow monolayer control in the chemical composition of the growing crystal. When two different semiconductors are grown into a single structure, the structure is called heterostructure. One such structure is superlattice in which two (or more) semiconductors A and B are grown alternately with thickness dA and dB, respectively (Figure below). GaAs-AlGaAs superlattice. On the left is a sequence of nearly thirty different layers, while on the right the individual atomic resolution is indicated.
Quantum Wells: It is possible to grow a heterostructure such that a narrow-bandgap material is surrounded by a large-bandgap material. An important example is the GaAs and the AlAs system shown below. Quantum wells are formed in the conduction and valence bands. Schematic of a quantum well structure formed in semiconductor heterostructure. Note there are two quantum wells, one for the conduction band and the other for the valence band.
An extremely important parameter in the quantum well problem is the bandedge discontinuity produced when two semiconductors are brought together. A part of the bandgap discontinuity (EgA-EgB) of two semiconductors A and B would appear in the conduction band and a part would appear in the valence band. The ratio of the two parts for GaAs and AlAs is • i.e. 60% to 65% of the direct bandgap difference between AlAs and GaAs is in the conduction band, and the remaining bandgap is in the valence band. • The electrons in the narrow-bandgap material are confined in the crystal growth direction (z direction in the figure). If the barriers (walls) of the well are very high, we can treat the barrier heights as infinite and the energy is given by Note the last two terms corresponding to freely moving electrons in x and y directions.
Examples • For a semiconductor alloy, In0.47Ga0.53As, what is the band gap? Can you explain why this material is useful for infrared applications? Note that the energy gaps for InAs and GaAs are 0.354 eV and 1.424 eV, respectively.
Homework P4.17 Problem 7, textbook, page 159. P4.18 In the GaAs/AlGaAs heterostructure, 60% of the bandgap discontinuity is in the conduction band of the narrow-gap material. Calculate the conduction band and valence band quantum well potentials for GaAs/Al03Ga0.7As/ Can we use the infinite barrier model for obtaining the confined energy levels? The bandgap difference between GaAs and Al03Ga0.7As is 0.374 eV.
