A novel coupled-quantum-well system is described in which the in-plane, anisotropic strain in successive well layers alternates between compression and tension. A polarization anisotropy in the interband optical matrix elements that arises due to anisotropic strain is reversed between the compressive and tensile cases. Hence, transitions associated with the different well layers have reversed polarization anisotropies. The structure of interest has great flexibility in the energies of successive interband transitions, and in the size of the anisotropy of the various transitions. The structure can be used in describing quantum-well properties, in optical multiplexing, and in devices such as modulators.
© 2002 Optical Society of America
It is well known that anisotropic, in-plane strain leads to a polarization-based, optical anisotropy in quantum wells.[1–7] This polarization anisotropy can be exploited in optoelectronic devices, such as modulators.[4,7,8] An important feature of the polarization anisotropy is that it is reversed between the cases of compressive and tensile, anisotropic strain.[3,9] Hence, a structure incorporating quantum wells that are under both anisotropic compression and tension would provide added degrees of flexibility and control beyond that from a structure with only one type of strain. In this paper we describe a coupled-quantum-well structure (CQWS) in which successive well layers are under either compressive or tensile, anisotropic, in-plane strain.[10,11] The CQWS provides a polarization-based optical anisotropy that varies with subband type and with the well in which the subband originates. Anisotropic strain such as required in the CQWS of interest can be obtained by various means, including packaging techniques[2–4,12,13] and micromachining.[14,15] In the following discussion, a micromachining approach is assumed which allows the relaxation of lattice-mismatch-induced strain along only one in-plane direction. Other approaches could also be used to achieve the desired strain.
The proposed CQWS makes use of principles applied in earlier devices, but provides greatly enhanced flexibility and potential applicability when compared to earlier work. Shen, et. al.,[3–5,7,8] Burak, et. al., and Gil, et. al.[10,11] have made significant contributions to basic quantum well physics and to the development of opto-electronic devices by exploiting a single type of anisotropic, in-plane strain, usually compressive, in multiple quantum wells. Huang, et. al.[1,2] made use of both compressive and tensile, anisotropic, in-plane strain in a single well layer. The structure proposed in this work also exploits both compressive and tensile, anisotropic strain. However, the different types of strain are isolated in different well layers in the CQWS. Thus, in the proposed structure, the added flexibility and control associated with simultaneously exploiting both types of anisotropic strain is combined with the ability to tailor the subband structure and amount of strain independently for each type of strain. The ability to independently control the subband structure and amount of strain for each type of strain is unique to the proposed structure.
The nature of the optical anisotropy in quantum wells under anisotropic strain depends on both the subband and strain types. For transitions associated with heavy-hole subbands, anisotropic, in-plane strain will tend to enhance (suppress) the interband, optical matrix element for light polarized along (normally to) the quantization direction defined by the anisotropic strain.[3,7,9] The polarization anisotropy is reversed for light-hole-subband transitions. The quantization direction is along the in-plane axis that is under compression relative to the other in-plane axis.[9,16] Additionally, tension is more effective, in general, at producing a polarization anisotropy than is compression.
2 System and Approach
The key features of the CQWS of interest are illustrated through an example, although implementations of the basic structure are limitless. The structure analyzed consists of well layers under compression, consisting of Ga0.412In0.588As, alternating with well layers under tension, consisting of Ga0.53In0.47As. The compressive well is fixed at a thickness of 26 angstroms, while the tensile well varies in thickness from 26 to 64 angstroms. All barrier layers are 21 angstroms thick and consist of Al0.49In0.51As. For this example CQWS, the substrate is assumed to be InP, which is lattice matched to the barrier layer. The pseudomorphic assumption is made, with the strain primarily in the well layers. With the convention that compressive strain is negative, lattice mismatch leads to a strain of -0.46% in the well under compression, and a strain of 0.35% in the tensile well, producing about the same optical anisotropy in subbands arising from both types of wells. The well under compression has a smaller bandgap than the well under tension. The valence bandedge offset is treated in the same way for both the well under tension and the barrier layers, with 0.334 of the bandgap difference associated with the valence subbands. The in-plane strain is assumed to be anisotropic in both types of wells, with the strain along the x-axis, EXX, set equal to the lattice mismatch induced strain, and the strain along the y-axis, EYY, equal to 0.3(EXX). The CQWS can be fabricated by traditional MBE or MOCVD techniques that are well documented in the literature. The desired strain anisotropy can be achieved by attaching, at other than room temperature, the CQWS to a material with a thermal expansion coefficient that differs from that of the CQWS, and then allowing the CQWS to return to room temperature.[2–4,12,13] Alternatively, the strain anisotropy can be obtained by micromachining techniques.[14,15]
The superlattice is modeled using a formalism which simultaneously calculates the conduction, heavy-hole, light-hole and spin-split-off bands of the bulk materials, with additional bands coupled in using k∙p perturbation theory.[17–22] Strain and the spin-orbit coupling are included between the explicitly treated bands. Matching at bulk material interfaces is achieved using the normal component of the current density matrix. The model allows for full band mixing, which is crucial since the optical-matrix-element, strain dependence relies strongly on band mixing.
3 Transition Energies
Calculations for the CQWS show that it displays the desired features. The total transition energies, ignoring exciton binding energies, for several of the lowest-energy, interband transitions are illustrated in Fig. 1. The transition energies are graphed as a function of the width of the well under tension, with the width of the compressive well held fixed. Since the coupled quantum wells are asymmetric, the states are associated primarily with one or the other of the two types of wells. Hence, the state heavy-hole 1, or HH1, is associated primarily with the well under compression for tensile-well widths up to about 55 angstroms, when HH1 in the tensile well replaces HH1 in the compressive well as the highest-energy, valence state. HH2 is associated primarily with the tensile well, until the subband crossing just described occurs. Both the light-hole 1, or LH1, subband and the conduction 1, or C1, subband are mixed between both well types for tensile-well widths up to about 35 angstroms, after which they are associated with the tensile well. In the same way, LH2 and C2 are mixed between both well types for very narrow tensile well widths, and are associated with the compressive well for tensile-well widths greater than 35 angstroms. While the transitions HH3-C1 and LH2-C1 are in the same energy range as those illustrated, they are not included since their interband, optical matrix elements are essentially zero for all cases considered.
4 Interband Optical Matrix Elements
The squared, optical matrix elements for transitions from heavy-hole subbands to conduction-band states are illustrated in Figs. 2 and 3. The results are graphed as a function of the width of the tensile well. The optical matrix elements for light polarized along the z-axis are essentially zero for the heavy-hole to conduction-band transitions.
The polarization anisotropies are as expected for heavy-hole to conduction transitions, with the optical matrix element enhanced along the quantization direction in the well with which the particular valence state is primarily associated. For example, the matrix element for the HH1-C1 transition is enhanced for light polarized along the x-axis when HH1 is associated with the compressive well. However, when HH1 shifts to the tensile well for large tensile-well widths, the polarization anisotropy reverses and the matrix element for light polarized along the y-axis is enhanced.
Three aspects of Figs. 2 and 3 are of note. First, for a tensile well width of about 35 angstroms, the matrix elements for the HH1-C1 and HH2-C1 transitions are approximately equal, but their anisotropies are reversed. Since the transitions are only separated by about 19 meV, and the polarization anisotropy is significant, one could make use of the double polarization anisotropy in an optical modulator or similar device. For example, such a system could be used for polarization-based optical multiplexing. Also of interest is the trend of the size of the optical matrix elements with tensile-well width. For transitions to both C1 and C2, as the tensile-well width increases the matrix element increases (decreases) for the valence subband associated with the same (opposite) well as the conduction subband. Finally, it should be noted that for a tensile-well width of about 55 angstroms, a heavy-hole subband crossing occurs, dramatically impacting the matrix elements. That is, for tensile-well widths greater than 55 angstroms, the tensile-well, first heavy-hole state replaces the compressive-well, first heavy-hole state as being highest in energy. For tensile well widths near that at which the heavy-hole subband crossing occurs, the degree of anisotropy and the size of the matrix elements change significantly. However, it can be seen that the overall trends for the size of the matrix elements are re-established after the region of the subband crossing.
The squared, optical matrix elements for the LH1-C1 transition are illustrated in Fig. 4. The mixing of the LH1 and C1 states between both types of wells for well widths below 35 angstroms leads to very small polarization anisotropies. This follows because the anisotropies are opposite in the compressive and tensile wells, and so the anisotropy from one type of well will tend to cancel that from the other type of well. As the tensile well becomes wider, and both the LH1 and C1 envelope functions become localized in the tensile well, the polarization anisotropy grows. Note that the matrix element for light polarized in the plane of the quantum well, but normally to the quantization direction, is enhanced, as is expected for a light-hole transition.[7,9,16]
In summary, we have described an asymmetric, CQWS. By making use of tensile and compressive, anisotropic, in-plane strain, the structure creates two sets of interband transitions, between which the optical polarization anisotropies are reversed. Because the two sets of transitions are associated with different well layers, the bulk material bandgap, and, hence, total transition energy, can be controlled independently for each set of transitions, which gives the structure both flexibility and functionality. By varying the CQWS parameters, such as lattice mismatch and well width, a variety of transition energies, matrix element strengths and degrees of anisotropy can be achieved. This system shows promise for use in polarization-based, optical modulators, and could provide the basis for polarization-based, optical multiplexing. With the great variety of semiconductor alloys available, various implementations of the CQWS structure proposed here could serve as the basis for optoelectronic devices that would operate from the near infrared, in the region of 1.5 microns, well into the visible.
The authors wish to acknowledge the support of the Office of Naval Research in carrying out this work.
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