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Galerkin Method for Calculating Valence-Band Wavefunctions in Quantum-Well Structures Using Exact Envelope Theory Gordon B. Morrison, Sean C. Woodworth, Huiling Wang, and Daniel T. Cassidy
Abstract—The ability to calculate accurately the valence-band structure in semiconductors is important in the design of quantum-well (QW) semiconductor devices. The Galerkin method for calculating accurate analytic approximations for multicomponent valence-band wavefunctions is computationally fast and efficient. The Galerkin method is, in fact, an improved version of an earlier Raleigh–Ritz-type variational method. In this paper, we remark that both the variational method and the more recently proposed Galerkin method are formulated such that they imply symmetrized boundary constraints at material interfaces, with the symmetrized nature of the constraints arising from neglecting the ordering of the operators. Burt’s exact envelope-function theory for semiconductor microstructures has been used to demonstrate, however, that the commonly used symmetrized boundary constraints for material interfaces are unphysical. We therefore present a modified version of the Galerkin method that implicitly assumes physically reasonable, exact envelope-function boundary constraints. Simulations show that the modified Galerkin method successfully produces physical, semi-analytic results that are consistent with exact envelope theory. Index Terms—Galerkin method, multiple-quantum-well lasers, semiconductor lasers, valence-band modeling.
I. INTRODUCTION
E
NGINEERS are designing increasingly sophisticated multiple-quantum-well (MQW) laser devices for specific applications such as telecommunications or absorption spectroscopy [1], [2]. The tuning of spectra in these new devices is critical in advanced semiconductor laser applications and depends heavily on the design of MQW active regions. Valence-band mixing in QWs and carrier tunneling between QWs must be considered to predict correctly the optical gain in MQW active region designs. Several methods have been used to solve the second-order coupled differential equations that are found in the valence-band effective mass Hamiltonian. Generally, these techniques are numerical and include the transfer matrix method and finite-element method [3]–[6]. These methods are computationally time-
consuming and somewhat complicated, and accuracy depends strongly on grid size [7]. Numerically defined wavefunction solutions also result in slower calculation of overlap integrals in optical gain calculations. The Galerkin method proposed by Trenado and Palmier [7] provides a semi-analytic method for determining valence-band wavefunctions in QW structures. The Galerkin method for analysis of valence-band mixing in QWs can be viewed as an extension of the variational Raleigh–Ritz approach that was used in earlier work [8], [9]. A key difference exists, however, between the variational approach and Trenado’s Galerkin formulation. This difference is in the choice of decoupled single-QW (SQW) solutions used for approximation of the Hamiltonian eigenfunctions. In this paper, we will demonstrate the superiority of the more recent Galerkin approach. All of the aforementioned techniques for dealing with valence-band mixing in MQW structures begin with a valence-band Hamiltonian. A common approach is to use the Luttinger–Kohn (LK) Hamiltonian [10] for homogeneous bulk material, but the abrupt changes in material parameters at heterojunction interfaces result in a non-Hermitian Hamiltonian [11]. Many arbitrarily chosen modifications to the LK Hamiltonian have been suggested to force Hermitian properties on the Hamiltonian [11]. The choice of Hamiltonian affects the nature of the boundary constraints, owing to the piecewise continuous nature of the Hamiltonian at material interfaces. In this paper, we demonstrate that the Galerkin method, though not explicitly employing boundary constraints, inherently assumes symmetrized boundary constraints. Symmetric boundary constraints have been shown by Foreman to be physically unreasonable [12]. We demonstrate an alternative implementation of the Galerkin method that intrinsically assumes boundary constraints that are consistent with Burt’s exact envelope-function theory [12], [13].
II. LK HAMILTONIAN Manuscript received September 9, 2003; revised November 24, 2003. This work was supported by the Ontario Photonics Consortium. G. B. Morrison was with the Department of Engineering Physics, McMaster University, Hamilton, Ontario, Canada. He is now with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106 USA. S. C. Woodworth, H. Wang, and D. T. Cassidy are with the Department of Engineering Physics, McMaster University, Hamilton, ON L8S 4L7, Canada (e-mail:
[email protected]). Digital Object Identifier 10.1109/JQE.2003.823024
In this paper, we will deal with the standard upper block 3 3 Broido–Sham [8] transformed LK Hamiltonian, with axial approximation and spin-orbit coupling [3]. In its usual form, the for the valence-band envelope wavefunctions Hamiltonian is written as
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(1)
MORRISON et al.: GALERKIN METHOD FOR CALCULATING VALENCE-BAND WAVEFUNCTIONS IN QW STRUCTURES
where is a scalar energy measured with respect to the band edge. The upper block 3 3 Hamiltonian is written as
(2) with
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are complex variables that make up the where the is a complex amplitude. The functions eigenvector and describe the envelope wavefunction components in the and directions for each of the Broido–Sham basis states where the , or ) indicates the HH, LH, and SO comsubscript ( ponents, respectively, of the envelope wavefunction. For ease of notation, from here on this paper will only work with the component (QW growth direction) of the wavefunction , and the multiplying coefficients will be omitted but assumed. In more complex situations, such as in application to QW structures, the Hamiltonian equation (1) is taken as piecewise continuous with boundary conditions for bound states (i.e., at ) and boundary constraints that require matching wavefunctions at material interfaces. A more general form of solution to the coupled differential equations in each of the materials must therefore be assumed, i.e.,
(3)
(5)
and the Luttinger parameters , the electron affinity , strain potentials and , the bandgap energy , and the spin-orbit split off energy are empirically determined or interpolated for various semiconductors [3]. Here we define as the potential energy for the holes at the top of the unstrained, bulk valence band. Energy is defined as decreasing for the holes in the valence band away from the band edge. The Hamiltonian of (2) is for the envelope of the basis states [3], [12] and is computed for a given value of the in-plane, parallel wave vector . The envelope wavefunction vector has three components, each representing the envelope for one of the Broido–Sham type upper Hamiltonian basis states. The original six-component basis of the LK Hamiltonian, , light-hole which is composed of heavy-hole , and split-off Bloch states, is changed to a three-component basis under the unitary transformation that block diagonalizes the LK Hamiltonian. The basis of the 3 3 Broido–Sham Hamiltonian retains the heavy-hole (HH), light-hole (LH), and split-off (SO) character for the Bloch wavefunctions since each new basis consists of a linear combination of the angular momentum states for a given HH, LH, or SO Bloch state [3]. Every eigenenergy will be associated with a separate wavefunction vector . As is common in eigenequation notation, the subscript label of in (1) is henceforth neglected. The term appears in (3) instead of the Hermitian operator in bulk material only, where it is reasonable to assume a particular solution [14] of the form
cannot necessarily be replaced with and the operator as in (2) (i.e., ) [15]. Note that, because of a small difference in choice of basis state definition, the form of the LK Hamiltonian used in this paper is slightly different from that found in the original paper [10]. These differences manifest themselves in the form of a factor of in front of certain terms in . III. GALERKIN VERSUS VARIATIONAL METHOD FOR A SINGLE QW The variational method used by Broido and Sham and Twardowski and Hermann was applied to a 2 2 Hamiltonian that neglects the spin-orbit split-off band [8], [9]. In this variational method, heavy hole and light hole decoupled SQW envelope functions are obtained by solving the equations that result from setting the off-diagonal terms in the Hamiltonian to zero. The decoupled equations are standard textbook QW eigenfunction equations [16], [17]. Solutions for the decoupled “basis” functions have evanescent tails in the barriers and look like typical waveguide solutions [7]. Throughout this paper, we will refer to solutions that are obtained by ignoring the coupling terms (i.e., the off-diagonal terms) in the Hamiltonian as “decoupled” solutions or functions. In the variational method, one assumes that multicomponent eigenfunctions of the envelope Hamiltonian can be expanded using decoupled QW envelope functions such that (6)
(4)
where and are decoupled heavy-hole and light-hole QW and envelope functions, respectively, and the coefficients
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define the contributions of the decoupled envelope functions to the overall envelope wavefunction. A key difference between this variational method and the Galerkin method proposed by Trenado et al. [7] is that, in the and are each assumed to be linear sums Galerkin method, of all of the decoupled envelope eigenfunctions, i.e., (7) The Galerkin method clearly introduces a larger selection of decoupled QW envelope functions for expansion of each component, or basis state, of the envelope wavefunction, thereby assuring a more accurate final solution [18]. In general, however, if there is more than one energy level for each of the decoupled solutions, the variational method can provide adequate solutions to the 2 2 Hamiltonian when is not too large. Fortuitously, Broido and Sham [8] and Twardowski and Hermann [9] were near zero. primarily interested in carriers with We have found that, if the variational formulation is extended to problems in which the 3 3 Hamiltonian is necessary, i.e.,
Fig. 1. Calculated valence subband structure for the lattice-matched 60- A Ga As–InP QW reported by [3]. The solid lines show the result using In the Galerkin method and the dotted line shows the variational method band diagram. For a comparison, the squares show the band structure calculation using the transfer matrix method. The variational method is found to underestimate the strength of the coupling between subbands compared to the Galerkin method.
(8) then the effects of coupling between the light-hole and split-off bands are significantly underestimated. Fig. 1 demonstrates this phenomenon and compares the results to those obtained by way of the Galerkin method, which assumes solutions of the form
(9) For reference, an additional band structure that has been calculated using a transfer matrix method [3]–[5] is included in the figure. Clearly, Trenado’s Galerkin approach is better able to approximate the coupling between the light-hole and split-off bands than the variational approach. The importance of using Trenado’s Galerkin formulation [7] to approximate the multicomponent valence-band wavefunction is better understood if one considers the form of a general solution to three coupled second-order differential equations. The decoupled envelope functions used in the Galerkin and variational methods are solutions to the decoupled, second-order effective mass equations, and therefore consist of a pair of exponentials with positive and negative wavenumbers. The general solution for the multicomponent wavefunction will then consist of six three-component eigenvectors, with six exponential
terms [14] as shown in (5). Thus, a required solution to the eigenvalue problem is better approximated by allowing each component of the wavefunction to be a mixture of all three types of decoupled functions. It is important to realize that eigenenergies obtained by the Galerkin method will necessarily be overestimates, not underestimates, due to the positive definite nature of the Hamiltonian. The lowest eigenenergies will be most accurate, with increasing error in the higher eigenenergy estimates [18]. For calculations regarding gain in QWs, the lower energy bound solutions are the most important. To increase the accuracy of higher eigenenergy estimates, one can attempt to include more decoupled functions, such as functions confined only by the cladding, as was suggested by Trenado and Palmier [7]. Naturally, the introduction of additional decoupled functions will result in the computation of additional, larger eigenenergies, which should be ignored as they will be grossly overestimated and will, of course, be far above the barrier potentials of the well. IV. GALERKIN METHOD AND IMPLIED BOUNDARY CONSTRAINTS The Galerkin method applied to MQW structures uses decoupled states from all wells. The decoupled states in each well are calculated without consideration of the effects of neighboring wells. A solution for an MQW structure which has wells is thus assumed to have the form given by (10), shown at the bottom of the page, i.e., each wavefunction is a linear sum
(10)
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of all heavy-hole (hh), light-hole (lh), and split-off (so) enve. The superscript inlope wavefunctions from all wells dicates that the decoupled QW envelope function is from the th well for the wells. For example, it is assumed that there decoupled heavy hole eigenfunctions for the th well. are The total number of basis envelope wavefunctions is . As described by Trenado and Palmier [7], the three-dimensional residual vector obtained by substituting the assumed solution into the differential equation (1) is
is positive definite. The decoupled QW envelope functions are real, so therefore a Hermitian matrix [19], [16] yields a matrix. Note that, for a real function, . symmetric The conjugate notation will therefore be dropped henceforth in this paper. The LK Hamiltonian for bulk materials is, in fact, Hermitian. However, when applied to a QW problem, the dependences of the Luttinger parameters introduce source terms when integrating across abrupt material interfaces and this yields an matrix. That is, asymmetric
(11)
(19)
Following the Galerkin method, an inner product of the residue with each of the decoupled states must equal zero, . Thus, one must solve the that is, simultaneous equations that create the matrix equation
(20) although for square integrable functions
(12)
(21)
such that
(22) (13)
where (14) where and are indices running from 1 to 3, and and are , thereby including all decoupled indices running from 1 to QW envelope functions. The matrix is written as (15)
as is easily proven using integration by parts, including boundary source terms, and making use of the fact that the envelope functions and their derivatives have values of zero at (i.e., square integrable functions). As is commonly found in the literature, to ensure Hermiticity in the Hamiltonian, all first-order operators are often written as [3], [20], [21] (23) such that the first-order terms in the (see the Appendix)
matrix can be written as
where (24)
(16) Note that the positioning of the wells with respect to each other is very important. If wells are further apart, the associated decoupled wavefunction overlap integrals in (14) and (16) will naturally be much smaller, indicating much less coupling. vector is comprised of all the weighting coeffiThe cients and is written as (17)
where is a material-dependent Luttinger parameter. Second-order operators are often written as (25) such that the second-order terms in the (see the Appendix)
matrix can written as
(26)
Rewriting (12) as (18) one can use standard numerical techniques to solve for all eigenenergies of the coupled wells and their associated eigenvectors . To obtain real positive eigenenergies using the Galerkin method, the Hamiltonian must be Hermitian such that (12)
With these new symmetrized operators, the Hermiticity of is maintained even under the dependence in material parameters where a discontinuous step in the Luttinger parameters occurs at the boundary between layers. These changes in differential operators are commonly used in the literature but are arbitrary and cannot be physically justified [12].
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The Galerkin method endeavors to satisfy, as well as is possible, the piecewise-continuous Hamiltonian equation with the , and will be trial solution (10). The coefficients of type found such that the differential equation is best satisfied at all points in space, and the Galerkin residue is zero. The integral of the left-hand side of the piecewise-continuous Hamiltonian equation (1) over a region containing a boundary should be continuous but kinked [11], which is the same as the right-hand side of the equation (i.e., the integral of a discontinuous function will produce a kinked function). Thus, integration over from to , where is the value of at the boundary and approaches zero, results in the well-known boundary constraints [12], [23] shown in (27) at the bottom of the page. These boundary constraints, which are especially important to transfer matrix methods for modeling valence-band mixing in QWs, implicitly assume the symmetrized form of the operators given in (23) and (25) and have been shown to be physically unreasonable [12], [23]. Unfortunately, they do apply to the wavefunction that is approximated using the Galerkin method described above. Thus, a modification to the Galerkin method is in order. It is worth reiterating here that the Galerkin method only requires decoupled functions that satisfy boundary conditions, for bound states. Boundary conspecifically, straints at interfaces are implicitly included in the piecewise continuous nature of the Hamiltonian. Although interface boundary constraints could potentially be used to improve suitability of the decoupled functions in the material interface regions (and therefore the quality of approximation in these regions), the additional constraints would be at the expense of reduced approximation quality everywhere else [18]. Wavefunctions and eigenenergies that would be obtained by the Galerkin method using decoupled function vectors determined by the additional constraints of (27) would have reduced accuracy compared with the decoupled function method that has been presented here.
portant [12]. In the Burt–Foreman formulation, one obtains (28), shown at the bottom of the page, where
V. EXACT ENVELOPE FORMULATION OF THE GALERKIN METHOD
for conjugate terms in which the operators are exchanged. Here the decoupled functions are square-integrable, the material paor is constant in the radial , or parallel, rameter direction, and the parallel components of all decoupled functions are chosen to have a wavevector of .
Foreman has used Burt’s exact envelope-function theory [13] to obtain a Hamiltonian in which operator ordering is very im-
(29) and all other symbols are as defined in (3). Note that the Hermitian conjugate symbol implies a transposition of the operators and , where . , and are related to the well-known LutThe terms tinger parameters using the equations [12]
(30) where is the azimuthal angle of . Starting with the Burt–Foreman version of the Hamiltonian, the Galerkin formulation for first derivatives with respect to in the matrix now appear as (31) or as (see the Appendix) (32)
(27)
(28)
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The symmetry of the matrix is ensured because the exact envelope function Hamiltonian is Hermitian, as is demonstrated by
(33) where . In the Burt–Foreman Hamiltonian (28), all of the off-diagonal first derivative operators (where represents the first derivative term in the elematrix results. ment of the Hamiltonian), and a symmetric This is, of course, in distinct contrast to (19). The Burt–Foreman boundary constraints are obtained using the same arguments as were used to obtain boundary constraints for the symmetrized Galerkin method [24]. Operator ordering, however, causes some terms to be discarded. That is,
(34) and because
Fig. 2. Calculated valence subband structure for the lattice-matched 60- A In Ga As–InP QW reported by [3]. The solid lines show the result using the Galerkin method with Foreman’s exact-envelope theory Hamiltonian from (28). The dotted lines show the subband structure obtained using the Galerkin method with the symmetrized Hamiltonian from (2). For a comparison, the squares show the resulting band structure obtained from the transfer matrix method using the Hamiltonian and boundary constraints derived from the exact envelope theory.
is 0 we have
(35) The derivative of the discontinuous material parameter at boundary is a delta function with area such that (36) whereas for the Hermitian conjugate
(37) Thus, the modified Galerkin formulation that is derived starting with the Burt–Foreman Hamiltonian implicitly results in a solution that satisfies the physically reasonable but asymmetric Burt–Foreman boundary constraints, i.e., see (38), shown at the bottom of the page. In Fig. 2, a comparison is made of band structures that were obtained using the symmetrized Galerkin method with band
structures that were obtained using the exact-envelope version of the Galerkin method. The valence subbands were computed Ga As QW with for a 60- lattice-matched In [3]. Substantial differences appear between the two methods at increasing values of . For a comparison, Fig. 2 includes the calculated band structure obtained using a transfer-matrix approach with the exact-envelope theory Hamiltonian from (28) and the Burt–Foreman boundary constraints from (38). The good agreement observed in Fig. 2 of the calculated band structures from the Galerkin and transfer matrix method demonstrates the validity of the Galerkin approach when working with the Burt–Foreman exact-envelope theory. The multicomponent valence-band envelope functions have been calculated using the Galerkin method with Foreman’s exact envelope theory. The results are shown in Figs. 3 and 4 for the first heavy-hole and light-hole subbands of the QW structure that was used in Figs. 1 and 2. The Galerkin method is shown to provide excellent agreement with the envelope functions computed using the transfer matrix method both (Fig. 3) and (Fig. 4). Fig. 4 shows the at nm where the computed envelope functions at valence-band structure is highly nonparabolic. The coupling between the HH, LH, and SO components of the valence-band envelope function are found to be accurately determined using the Galerkin method. Moreover, the Galerkin method provides an analytic form for the envelope functions compared to the strictly numerical results for the envelope functions obtained from the transfer matrix method. The accurate determination of the envelope functions for the conduction and valence subband
(38)
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Fig. 3. Comparison of the band edge (k = 0) valence-band envelope functions calculated using the Galerkin method and transfer matrix method with Foreman’s exact envelope theory. The amplitude of the three-component envelope function is compared for the first heavy-hole and light-hole subbands A lattice-matched In Ga As QW with x = 0:468. The solid lines for a 60- show the Galerkin solution, and the transfer matrix method solution is shown using squares (HH component), triangles (LH component), and circles (SO component). The heavy-hole subband is completely uncoupled at k = 0, while the light-hole subband contains a mixture of light-hole and split-off components.
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Fig. 5. Calculated zone-center effective mass for the first two heavy-hole Ga As QW subbands. The effective masses were calculated using the In with x = 0:468 with InP barriers reported by [3]. The solid lines show the effective mass calculated using the Galerkin method with Foreman’s exact envelope theory Hamiltonian from (28). The dotted lines show the effective mass obtained using the Galerkin method with the symmetrized Hamiltonian of (2).
very sensitive to the QW width. This is the result of using the unphysical symmetrization procedure [12]. VI. CONCLUSION
Fig. 4. Comparison of the valence-band envelope functions calculated at k = 0:2 nm using the Galerkin method and transfer matrix method with Foreman’s exact envelope theory. The amplitude of the three-component envelope function is compared for the first heavy-hole and light-hole subbands for the QW structure of Fig. 3. The labeling of the envelope functions as heavy-hole (HH) and light-hole (LH) reflects the band edge (k = 0) solutions from Fig. 2, and we have chosen to maintain the same name designation for the solutions at k > 0. The solid lines show the Galerkin solution, and the transfer matrix method solution is shown using squares (HH component), triangles (LH component), and circles (SO component). Significant mixing between HH and LH components of the envelope functions are observed for the two subbands shown.
energies is important for the calculation of the gain spectrum of the QW structure [20]. The Galerkin method will therefore provide a useful means for calculating gain in QWs. effective masses Fig. 5 compares the zone-center calculated with the Galerkin method for the first two heavy-hole subbands as a function of QW width. Results are shown for the symmetrized Hamiltonian and the exact envelope Hamiltonian. The profile for the heavy-hole effective mass predicted by the symmetrized version of the Galerkin method is observed to be
The Galerkin method, although very similar mathematically to earlier Raleigh–Ritz variational approaches, is better able to approximate multicomponent valence-band wavefunctions in MQWs. This is because, in contrast to the Raleigh–Ritz-type variational method, the Galerkin method allows each component of the wavefunction to be a linear combination of all heavy-hole, light-hole, and split-off decoupled functions. Both methods implicitly assume symmetrized boundary constraints because both techniques are implemented using a Hamiltonian that must be symmetrized to preserve Hermitian properties. We have demonstrated a different implementation of the Galerkin method that is in agreement with the Burt–Foreman exact envelope-function boundary constraints. By comparison with the transfer-matrix model, we have demonstrated the accuracy of our exact-envelope formulation of the Galerkin approach. It is a fast, computationally efficient, physically reasonable, and accurate method for obtaining analytic approximations to MQW valence-band structures. APPENDIX Starting with the general form of the symmetrized first order differential operator, we can manipulate the inner products found in the Galerkin method as follows:
(A1)
MORRISON et al.: GALERKIN METHOD FOR CALCULATING VALENCE-BAND WAVEFUNCTIONS IN QW STRUCTURES
Starting with the general form of the symmetrized second-order differential operator, we can manipulate, using integration by parts, including source terms for integrating across the interface, and the boundary conditions for decoupled wavefunctions , the inner products in the Galerkin method as follows:
(A2)
Starting with the general form of the Burt–Foreman first-order differential operator, we can manipulate the inner products in the Galerkin method as follows:
(A3)
or, in the case where operators are reversed,
(A4)
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REFERENCES [1] D. M. Adams, C. Gamache, R. Finlay, M. Cyr, K. M. Burt, J. Evans, E. Jamroz, S. Wallace, I. Woods, L. Doran, P. Ayliffe, D. Goodchild, and C. Rogers, “Module-packaged tunable laser and wavelength locker delivering 40 mW of fiber-coupled power on 34 channels,” Electron. Lett., vol. 37, pp. 691–693, 2001. [2] S. C. Woodworth, D. T. Cassidy, and M. J. Hamp, “Experimental analysis of a broadly tunable InGaAsP laser with compositionally varied quatum wells,” IEEE J. Quantum Electron., vol. 39, pp. 426–430, Mar. 2003. [3] C. Y. Chao and S. L. Chuang, “Spin-orbit-coupling effects on the valence-band structure of strained semiconductor quantum wells,” Phys. Rev. B, vol. 46, pp. 4110–4122, 1992. [4] S. L. Chuang, “Efficient band-structure calculations of strained quantum wells,” Phys. Rev. B, vol. 43, pp. 9649–9661, 1991. [5] L. C. Andreani, A. Pasquarello, and F. Bassani, “Hole subbands in strained GaAs-Ga1-xAlxAs quantum wells: Exact solution of the effective-mass equation,” Phys. Rev. B, vol. 43, pp. 5887–5894, 1987. [6] G. Debaiseux, G. Herve-Gruyer, M. Filoche, S. Bouchoule, and J. F. Palmier, “Self-consistent 1-D solutionof multiquantum-well laser equations,” Opt. Electron., vol. 29, Apr. 1997. [7] N. Trenado and J. F. Palmier, “An efficient and accurate method for gain spectrum calculations in nonidentical multi-quantum-wells,” IEEE J. Quantum Electron., vol. 38, pp. 495–499, May 2002. [8] D. A. Broido and L. J. Sham, “Effective masses of holes at GaAs-AlGaAs heterojunctions,” Phys. Rev. B, vol. 31, pp. 888–892, 1985. [9] A. Twardowski and C. Hermann, “Variational calculation of polarization of quantum-wll photoluminescence,” Phys. Rev. B, vol. 35, pp. 8144–8153, 1987. [10] J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev., vol. 97, pp. 869–883, 1955. [11] M. G. Burt, “The justification for applying the effective-mass approximation to microstructures,” J. Phys.: Condens. Matter, vol. 4, pp. 6651–6690, 1992. [12] B. R. Foreman, “Effective-mass Hamiltonian and boundary conditions for the valence bands of semiconductor microstructures,” Phys. Rev. B, vol. 48, pp. 4964–4967, 1993. [13] M. G. Burt, “Fundamentals of envelope function theory for electronic states and photonic modes in nanostructures,” J. Phys.: Condens. Matter, vol. 11, pp. R53–R83, 1999. [14] M. R. Spiegel, Applied Differential Equations, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1981, pp. 502–527. [15] R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics. Don Mills, Canada: Addison-Wesley, 1963, vol. 1, ch. 23-1. [16] S. Gasiorowicz, Quantum Physics, 2nd ed. New York: Wiley, 1996. [17] R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Paricles, 2nd ed. Toronto, Canada: Wiley, 1974. [18] C. A. J. Fletcher, Computational Galerkin Methods. New York: Springer-Verlag, 1984. [19] B. A. Foreman, “Elimination of spurious solutions from eight-band k 1 p theory,” Phys. Rev. B, vol. 56, pp. R12748–R12751, 1997. [20] S. L. Chuang, Physics of Optoelectronic Devices. New York: Wiley, 1995, p. 183. [21] R. Eppenga, M. F. H. Schurmans, and S. Colak, “New k 1 p theory Al As-type quantum wells,” Phys. Rev. B, vol. 36, pp. for GaAs/Ga 1554–1564, 1987. [22] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures. Les Ulis, France: Wiley, 1988. [23] S. Ekbote, M. Cahay, and K. Roenker, “Tunneling properties of holes across abrupt heterostructures using Burt’s envelope function theory,” J. Appl. Phys., vol. 86, pp. 5650–5655, 1999. [24] R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics. Don Mills, Canada: Addison-Wesley, 1963, vol. 2, ch. 33-3.
Gordon B. Morrison received the B.A.Sc. (with honors) degree in engineering physics from Simon Fraser University, Vancouver, BC, Canada, in 1997 and the Ph.D. degree in engineering physics from McMaster University, Hamilton, ON, Canada, in 2002. He continued at McMaster University for eight months as a Postdoctoral Fellow. His graduate research included modeling the spectra of distributed feedback lasers and fitting the model to data for parameter extraction. Since then, he has been interested in design considerations for broadly tunable lasers. In June 2003, he joined the Department of Electrical and Computer Engineering, University of California, Santa Barbara, as a Visiting Assistant Research Engineer.
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Sean C. Woodworth received the B.Sc. degree in physics from Mount Allison Univeristy, Sackville, NB, Canada, in 1996 and the M.Sc. degree in physics from McMaster University, Hamilton, ON, Canada, in 1998, where he is currently working toward the Ph.D. degree in engineering physics. His research centers on the use of wide-gain-profile InGaAsP–InP lasers for molecular and chemical analysis, as well as the design of broadly tunable external-cavity diode laser systems.
Huiling Wang received the B.Eng. degree and M.Eng. degree in electronic engineering from Xi’an Jiaotong University, Xi’an, China, in 1994 and 1997, respectively. She is currently working toward the Ph.D. degree in engineering physics at McMaster University, Hamilton, ON, Canada. Her research concentrates on the study of optical gain and uneven carrier distribution in multiquantum-well lasers for the application as widely tunable laser sources.
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Daniel T. Cassidy received the B.Eng. degree in engineering physics from McMaster University, Hamilton, ON, Canada, the M.A.Sc. (Eng.) degree in electrical engineering from Queen’s University in Kingston, ON Canada, and the Ph.D. degree in physics from McMaster University. He is a Professor of Engineering Physics with McMaster University. His research interests are in the design, fabrication, and use of broadly tunable lasers in external cavities, the spectral properties of semiconductor diode lasers, and the characterization of materials and devices by analysis of the degree of polarization of luminescence. Dr. Cassidy is a member of the Canadian Association of Physicists, the Optical Society of America, the IEEE Lasers and Electro-Optics Society, and the Professional Engineers of Ontario.
