Comparison of Effective Potential Method and Rayleigh-Ritz Method for the Calculation of Energy Levels of Quantum Wires

Transcription

1 Journal of the Korean Physical Society, Vol. 34, No., April 1999, pp. S36 S41 Comparison of Effective Potential Method Rayleigh-Ritz Method for the Calculation of Energy Levels of Quantum Wires S. Y. Shim, Y. K. Lee, D. C. Kim K. H. Yoo Department of Physics Research Institute for Basic Sciences, Kyung Hee University, Seoul M. Jung T. W. Kim Department of Physics, Kwangwoon University, Seoul We calculated the electronic energy levels of quantum wires with rectangular crosssection by two methods, effective potential method Rayleigh-Ritz method. We compared the performance of these two methods by calculating the conduction b energy levels of GaAs/Al 0.3Ga 0.7As quantum wires. The results of the two methods agree well when the energy eigenvalues measured from the bedge of the well material are much less than the bedge energy difference between the well barrier materials. This observation is consistent with the expectation that the effective potential method gives accurate eigenvalues for well confined states. Also the results agree better when the aspect ratio of a quantum wire is larger. I. INTRODUCTION Recently quantum wires have attracted much attention because of their unusual physical properties associated with the reduced dimensionality [1] because of their industrial applications such as quantum wire lasers [2]. Calculation of the energy levels is the first step to underst the electronic properties of the quantum wires. Various methods have been used for the energy level calculations of quantum wires of various crosssectional shapes; rectangular [3 7], cylindrical [8], V-shaped [9 11], T-shaped [12], bent [13,14] so on. In this paper, we take two of the most popular methods, which we refer as effective potential method EPM) [3] Rayleigh-Ritz method RRM) [4] for rectangular quantum wires, elaborate the formalism, compare their performance by performing numerical calculations. In principle RRM should give the accurate results, even though in practice the accuracy depends on the choice of culculational parameters such as N L introduced later. EPM is an approximate formalism that reduces a 2-dimensional differential equation to two 1-dimensional equations. It enjoys the simplicity at the cost of accuracy. The aim of this paper is to determine quantitatively when the simple EPM gives sufficiently good results. We consider a quantum wire with a rectangular crosssection as shown in Fig. 1. In the one-b envelope function approximation [15], the Schrodinger equation for an electron is written as ) ] [ h2 1 2m o m x, y) + V x, y) Ψx, y) = EΨx, y), 1) where m o is the free electron mass, m x, y) V x, y) are the effective mass of the electron the heterojunction potential, respectively, in the well barrier regions. They are given by Fig. 1. Schematic diagram of a quantum wire with a rectangular crosssection. The width height of the well region are w x w y, respectively, the barrier height is V o. The barrier region is shaded, the hatched regions give errors in EPM. -S36-

2 Comparison of Effective Potential Method Rayleigh-Ritz Method for S. Y. SHIM et al. m mw if d x, y) = x < x < d x + w x d y < y < d y + w y, otherwise, 0 if dx < x < d V x, y) = x + w x d y < y < d y + w y, V o otherwise. -S37-2) 3) Here w x w y are the width height of the quantum wire, respectively, V o is the b offset, d x d y are arbitrary positions introduced for the convenience of calculation. The boundary condition is that Ψ becomes zero as x or y goes to infinity, the interface condition at the interface of the well barrier is that both Ψ 1 m Ψ are continuous. In EPM, the 2-dimensional quantum wire problem is reduced to two 1-dimensional quantum well QW) problems [3]. In RRM, the wavefunction is exped in terms of e waves [4]. Mathematically Eq. 1) is analogous [16] to the 2-dimensional waveguide problem for the calculation of the effective refractive index [17], EPM RRM correspond to the effective index method [17] the Rayleigh-Ritz method [18], respectively, in the context of waveguide. In RRM, we introduce the calculational parameters; number of e waves used in the expansion N x in Section IV, length of the region where Ψ is non- zero L x ). Since the accuracy of the calculation depends on these parameters, a wise choice of them is important. In Section II, we first apply RRM to a QW problem [19] in which we know the exact solution, find an empirical way to choose N L. In Section III IV, we elaborate the formalism of EPM RRM for quantum wires, respectively. In Section V, we apply the two methods to GaAs/Al 0.3 Ga 0.7 As quantum wires, discuss their performance by comparing the numerical results. Finally we draw conclusions in Section VI. II. RAYLEIGH-RITZ METHOD FOR QUANTUM WELLS The Schrödinger equation for a quantum well is ) ] [ h2 d 1 d 2m o dy m + V y y) χy) = ɛχy), 4) yy) dy Fig. 2. Contour plot of error in RRM energy RRM energy - exact eigenenergy) of a quantum well as a function. Q values corresponding to is shown at the top. The quantum well width is 100 Å = The numbers on the contour lines are the error in mev. We chose Q=100, increased along the dotted vertical line. Fig. 3. EPM energy-rrm energy) vs RRM energy for two quantum wires. The triangles are for w x=100 Å the filled circles are for w x=300 Å. = was used for both wires.

3 -S38- Journal of the Korean Physical Society, Vol. 34, No., April 1999 Table. 1. Difference in the energies in mev) calculated by the effective potential method the Rayleigh-Ritz method for the ground state of 10 different quantum wires. w x 100 Å 200 Å 300 Å 400 Å 500 Å = case = case where m yy) = mw if d y < y < d y + w y, otherwise, 5) V y y) = 0 if dy < y < d y + w y, V o otherwise. 6) In RRM [19] the wavefunction is exped in terms of e waves ) 2 ny πy χy) = a ny. 7) n y=1 In Eq. 7), it is assumed that χ becomes zero at y = 0 y = w y + 2d y, a finite number of e waves are included. If we substitute Eq. 7) into Eq. 4), multiply 2 myπy ) integrate over the interval [0, ], we get the following set of coupled linear Fig. 4. EPM energy-rrm energy) vs RRM energy for two quantum wires. The triangles are for = the filled circles are for = = m W. w x is 300 Å for both wires. equations for a ny, n y=1 where a ny = ɛ a my, 8) = h2 2m o ) 2 [ my π 1 +V o [ 1 w y 1 w ) y + 1 ) + 1 m y π cosm yπ) w y m W m W )] my πw y if m y = n y, ) 1 m y π cosm y π) )] my πw y 9) = h2 1 1 ) ) [ ) ) my n y π 1 my + n y )π my + n y )πw y m o m W L 2 cos y m y + n y 2 2 ) )] 1 my n y )π my n y )πw y + cos m y n y V [ ) ) o 1 my + n y )π my + n y )πw y cos π m y + n y 2 2 ) )] 1 my n y )π my n y )πw y cos if m y n y. m y n y )

4 Comparison of Effective Potential Method Rayleigh-Ritz Method for S. Y. SHIM et al. -S39- Eq. 8) is a typical eigenvalue problem, the energy level ɛ of the QW is the eigenvalue of the matrix. Since RRM results depend on, a wise choice of them is important. Since is the extent of the wavefunction, a suitable value of will depend on the eigenenergy. The wavefuction of a state with eigenenergy ɛ n decays in the barrier in a 2m form exp[± B m ov o ɛ n) y y h 2 o )]. We assume that the wavefunction is zero when this factor becomes 1/Q, get the following equation for, h 2 = w y + 2ln Q) 2m o V o ɛ n ). 11) In order to avoid the eigenenergy dependence, we use the parameter Q instead of. In order to find a good empirical way of choog Q, we performed numerical calculations for conduction b levels of GaAs/Al 0.3 Ga 0.7 As QWs. For input parameters, V o =0.374 ev, m W =0.067 two values = were used [20], w y was varied from 100 Å to 500 Å by 100 Å step. The RRM energies were compared with the exact eigenenergies calculated by the transfer matrix method TMM) [21,22]. An error in RRM energy refers to RRM energy - TMM energy). Typical results are shown in Fig. 2, where we draw a contour plot of the error in RRM energy as a function of. Fig. 2 is the results for the first excited state of 100 Å wide QW with = We see that the error is positive, i.e., RRM gives a larger eigenvalue than the exact one. As expected, the error goes to zero when both Q) becomes large. But, we note that, for a given allowable error, there is an optimum requiring the least ) value of Q. In this paper we aim at an error of 0.1 mev, which is roughly the typical experimental resolution. We see in Fig. 2 that for this error Q 20 is the optimum value. Unfortunately the value of optimum Q varies a little with varying well width eigenstates. As can be seen in Fig. 2, if we choose Q less than the optimum value, needed to achieve the given accuracy increases very rapidly; therefore it would be safe to choose Q to be a little larger than the typical optimum value. Therefore we propose the following procedure to achieve an accuracy of 0.1 mev. First we choose corresponding to Q=100. If the energy eigenvalue corresponding to is ɛ ), ɛ ) is monotonically decreases with increag along the vertical dotted line in Fig. 2). We increase by 2 The wavefunction is either even or odd function with respect to y = /2.) until both ɛ 4) ɛ 2) ɛ 2) ɛ ) are less than mev. Ug this procedure, we were always able to achieve the accuracy of 0.1 mev. III. EFFECTIVE POTENTIAL METHOD In this Section, we describe the effective potential method of a quantum wire described by Eq. 1). Assuming w x w y, we divide the potential V x, y) the inverse of the effective mass 1/m x, y) into two parts. For the potential 0 if dy < y < d V y y) = y + w y, 12) V o otherwise, wx, y) = V x, y) V y y). 13) Also we have for the effective mass 1/m 1/mW if d yy) = y < y < d y + w y, 1/ otherwise, 14) 1/µx, y) = 1/m x, y) 1/m yy). 15) If we neglect wx, y) 1/µx, y) in Eq. 1), Eq. 1) is reduced to Eq. 4) which is for a QW. We first solve for this QW, let ɛ n χ n y) be the n-th eigenenergy the normalized wavefunction of the QW. For the original quantum wire, we exp the wavefunction of Eq. 1) in terms of χ n y), Ψx, y) = n α n x)χ n y). 16) After we substitute this equation into Eq. 1), multiply χ my) integrate in the y-direction, we get the following set of equations for a n x) n [ h2 2m o x h2 2m o ) dyχ 1 my) m x, y) χ ny) x + dyχ my) y 1 µx, y) χ n y) y ) dyχ my)wx, y)χ n y) )] α n x) = E ɛ n )α m x). In EPM, we simplify these coupled equations to decoupled ones by taking only the n = m term. The simplified equation for a n x) is [ ) ] h2 d 1 d + Vn eff x) α n x) = E ɛ n )α n x), 18) 2m o dx x) dx m eff n 17)

5 -S40- Journal of the Korean Physical Society, Vol. 34, No., April 1999 which is another QW problem. If Pn W = d y+w y d y χ n y) 2 dy, i.e., the probability that an electron in χ n state resides in the well, Pn B = 1 Pn W, the effective mass m eff n the effective potential Vn eff are given by 1 m eff n x) = P W n m W + P B n if d x < x < d x + w x, 1 otherwise, 19) V eff n x) = 0 if dx < x < d x + w x, P B n V o h2 2m o 1 1 m W ) χ ny) d2 χ ny) dy 2 dy otherwise. 20) Since 1/µx, y) is assumed to be small in EPM, we may replace 1/m y y) in Eq. 4) by Pn W /m W + Pn B / ) solve for d 2 χ n /dy 2. If we substitute the resulting d 2 χ n /dy 2 in Eq. 17), the effective barrier height is simplified to Pn B m V o + B m W Pn W +Pn Bm )V o ɛ W n )Pn B. We note that the effective mass the effective potential in Eq. 18) depends on the subb index n of the QW in the y-direction. IV. RAYLEIGH-RITZ METHOD FOR QUANTUM WIRES RRM for quantum wire may be obtained by a straightforward extension of RRM for QW described in Section II. The wavefunction is exped as Ψx, y) = N x n x=1 n y=1 a nx,n y n x, n y, 21) where n x, n y = 2/L x n x πx/l x ) 2/ n y πy/ ), the coupled equations for a nx,n y are given by N x n x=1 n y=1 M mx,m y),n x,n y)a nx,n y = Ea mx,m y. 22) The matrix element M mx,m y),n x,n y) = m x, m y H n x, n y is similar to Eqs. 9) 10), but is too long to be written here. If we reorder the index of M a such as m = m x + m y m=1, 2,, N x ), Eq. 22) becomes a typical matrix diagonalization problem. V. NUMERICAL RESULTS AND DISCUSSION In order to compare the performance of EPM RRM, we calculated the conduction b levels of GaAs/Al 0.3 Ga 0.7 As quantum wires. V o =0.374 ev, m W =0.067 two values of = were used [20]. The wire height w y was fixed as 100 Å, while the wire width w x was varied from 100 Å to 500 Å by 100 Å step. We first calculated the eigenvalues ug the simple EPM. For RRM, we used the similar procedure to the case of QW with following modifications. For the energy eigenvalues needed to calculate L Eq. 22)), we used the result of EPM. For N i i = x, y), we used an empirical formula N i =5 number of nodes in the i-direction)+n, increased N by two. Without knowing the exact eigenvalues, we do not know the error exactly, but we guess, from the experience of QW the test ug sufficiently large N, that the error of RRM is about 0.1 mev. In Fig. 3, we plot EPM energy-rrm energy) versus RRM energy for the case w x =100 Å 300 Å with = If the energy eigenvalues are less than 0.5 V o 190 mev), the difference in energy between the two methods is less than 0.5 mev. Especially the difference for the ground state is very small. In Table 1, we list the difference for the ground state of 10 different quantum wires, we found that the difference is less than 0.1 mev in all the 10 cases. Therefore if one is interested only in the ground state, the simple EPM is good enough. As the eigenvalues become larger, EPM begins to yield a substantially larger eigenvalues than those by RRM. In the waveguide theory, the error in the effective index method is known to originate from the region corresponding to, in the quantum wire version, the hatched regions in Fig. 1 [23,24]. In analogy, we guess that, if the wavefunction is well confined in the well so that the wavefunction is negligible in the hatched regions, EPM would give an accurate result. The results in Fig. 3 are consistent with this expectation. Fig. 3 also shows that, for the same magnitude of eigenvalues, EPM gives less error in w x =300 Å case than in 100 Å case. The fact that EPM is better for a larger aspect ratio w x /w y ) is true of other values of w x. In Fig. 4, we compare the case = m W = the case = calculated for w x =300 Å. The agreements between EPM RRM are better in the case = m W = This might be associated with the error in the calculation of the second term in Eq.

6 19), which vanishes when = m W. VI. CONCLUSIONS We elaborated the details of the effective potential method the Rayleigh-Ritz method to calculate the energy levels of quantum wires with rectangular crosssection. Especially we discussed in detail the procedure to choose N L in RRM. In order to compare two methods, we numerically calculated the conduction b levels of GaAs/Al 0.3 Ga 0.7 As quantum wires. We found that, for states whose energy is less than half of barrier height, the results by the two methods agree well. Especially the ground state energies agree within 0.1 mev for all the 10 cases examined, showing that for the ground state the simple EPM is good enough. EPM gives larger eigenvalues for the higher lying states. The agreement between EPM RRM is better when the aspect ratio of a quantum wire is larger. ACKNOWLEDGMENTS This work was supported in 1998 by the Korea Science Engineering Foundation through the Center for the B Gap Modification of Exotic Materials. REFERENCES [1] H. Sakaki, Jpn. J. Phys. 19, ). [2] E. Kapon, Quantum Well Lasers, edited by P. S. Zory Academic, New York, 1993), p. 461 [3] J. A. Brum G. Bastard, Superlattices Microstructures 4, ). [4] D. Gershoni, H. Temkin, G. J. Dolan, J. Dunsmuir, S. N. G. Chu M. B. Panish, Appl. Phys. Lett. 53, ). [5] J. Shertzer L. R. Ram-Mohan, Phys. Rev. B41, ). [6] G. A. Baraff D. Gershoni, Phys. Rev. B43, ). [7] U. Bockelmann G. Bastard, Phys. Rev. B45, ). [8] P. C. Sercel K. J. Vahala, Phys. Rev. B42, ). [9] A. Saár, S. Calderon, A. Givant, O. Ben-Shalom, E. Kapon C. Caneau, Phys. Rev. B54, ). [10] O. Stier D. Bimberg, Phys. Rev. B55, ). [11] S. Gangopadhyay B. R. Nag, J. Appl. Phys. 81, ). [12] W. Langbein, H. Gislason J. M. Hvam, Phys. Rev. B54, ). [13] T. Kerkhoven, M. W. Raschke U. Ravaioli, J. Appl. Phys. 74, ). [14] J. P. Carini, J. T. Londergan, D. P. Murdock, D. Trinkle C. S. Yung, Phys. Rev. B55, ). [15] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures Halsted, New York, 1988), Chap. 3. [16] D. Campi, M. Meliga A. Pisoni, IEEE J. Quantum Electronics 30, ). [17] H. Nishihara, M. Haruna T. Suhara, Optical Integrated Circuits McGraw-Hill, New York, 1989), Chap. 2. [18] R. E. Collin, Field Theory of Guided Waves, 2nd ed. IEEE Press, New York, 1991), Chap. 6. [19] A. Abou-Elnour K. Schuenemann, J. Appl. Phys. 74, ). [20] S. Adachi, J. Appl. Phys. 58, R1 1985). [21] B. Chen, M. Lazzouni L. R. Ram-Mohan, Phys. Rev. B45, ). [22] M. Jung, T. H. Park, T. W. Kim, J. Cho, K. H. Yoo K-H. Yoo, Sae mulli New Phys.) 34, ). [23] A. Kumar, D. F. Clark B. Culshaw, Optics Lett. 13, ). [24] K. S. Chiang, Optics Lett. 16, ).