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Stark effect of electrons in semiconducting rectangular quantum boxes
ARTICLE IN PRESS
Microelectronics Journal 39 (2008) 786–791 www.elsevier.com/locate/mejo
Stark effect of electrons in semiconducting rectangular quantum boxes Guozhu Weia,b,, Sheng Wanga, Guangyu Yia a
b
College of Sciences, Northeastern University, Shenyang 110004, China International Center for Material Physics, Academia Sinica, Shenyang 110015, China Received 27 September 2007; accepted 15 December 2007 Available online 31 January 2008
Abstract The Stark shift of the electronic energy levels in semiconducting rectangular quantum boxes with different sizes is investigated by the use of variational solutions to the effective-mass approximation for electric fields of various orientations with respect to the center axis of the box. The asymptotic expansions of the Stark shift are given in the limits of low and high fields, respectively; they clearly indicate that the Stark shift is a quadratic function of the electric field for low electric fields and is an approximate linear function of the electric field for high electric fields. Likewise, our results also show that the largest Stark shift is obtained for the field directed along the diagonal in a cubic box, and is found for the low field directed along a side of the box and for the high field along the diagonal in a rectangular one. The large Stark shift of the electron and hole trapped in a quantum box leads to an obvious reduction of the interband recombination. r 2008 Elsevier Ltd. All rights reserved. Keywords: Quantum boxes; Stark effect
1. Introduction In recent several decades, man-made low-dimensional solids yield challenges in microstructure materials science. It is important for fundamental physics and for the development of device concepts to excite electronic properties due to size quantization. The study of energy shifts of particles in semiconductor nanostructures of different geometries turns out to be a very important task since the shifts are associated with the spatial separation between electrons and holes in these structures which induces a reduction of interband recombination. It is useful to understand quantum lasers. Consequently, the quantum confined Stark effect (QCSE) of low-dimensional system has attracted considerable attention recently. There has been much work on Stark effect in quantum wells (QWs), quantum wires (QWRs), quantum dots (QDs) [1–19]. Ham and Spector [17] have investigated the effect of an electric field on the electronic energy levels in a semiconducting Corresponding author at: College of Sciences, Northeastern University, Shenyang 110004, China. E-mail address: [email protected] (G. Wei).
0026-2692/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.12.012
quantum disk by the use of variational solutions and found that the dependence of the Stark shift on the sizes of the disks is related to applied electric field directions with respect to disk axis. Spector and Lee [18] have calculated the effect of a transverse electric field on the ground and the first few excited states of the electrons confined in a cubical quantum box by using the expansion of the wave functions method and used these results to examine the effect of the electric field on the intersubband optical absorption in such quantum boxes. They found that the application of the electric field leads to a Stark shift of the electron energies, which is quadratic in the field at low fields but becomes almost linear in the field at high fields. There has been less work concerning the Stark effect in rectangular quantum boxes. Li and Xia [19] have calculated the Stark effect of the energy of a hydrogenic donor impurity in a rectangular parallelepiped-shaped quantum dot in the framework of effective-mass envelope-function theory using the plane wave basis. In this paper, we want to investigate the Stark shift of the energy levels in rectangular quantum boxes depends upon different sizes of the box for electric fields of various orientations with respect to the axis of the box by use of
ARTICLE IN PRESS G. Wei et al. / Microelectronics Journal 39 (2008) 786–791
variational calculation method, and emphasize how to obtain the largest Stark shift of the particle confined in a rectangular quantum box through changing the direction of the electric field. It is necessary to obtain the large Stark shift which leads to the high spatial separation of the electrons and holes inducing the reduction of interband recombination in these quantum structures. These results are relevant to any optical processes, such as emission and absorption. The paper is organized as follows. In Section 2, we describe the model for the rectangular quantum boxes and the variational method used to determine the Stark shift of the electric energies. In Section 3, the results are presented and discussed. In Section 4, conclusions are presented.
We take our variational wave function as px py pz cos cos Cðx; y; zÞ ¼ Nða; b; gÞ cos L W H ax gz by exp exp exp , L W H
* * ~ p2 * H¼ q F r þV c ð r Þ, (1) 2m ~ is the electric field, m* where ~ p is the carrier momentum, F and q are the carrier effective mass and charge, respec* tively, and r is the position vector of the electron. The confining potential is given by
V c ðx; y; zÞ ( 0 ¼ 1
_2 a2 _2 b2 _2 g 2 þ þ 2 2 2m L 2m W 2m H 2 a 1 coth a þ qLF sin y cos j 2 þ a þ p2 2a 2 b 1 coth b þ qWF sin y sin j 2 þ 2 b þ p2 2b g 1 coth g þ qHF cos y 2 þ , ð5Þ g þ p2 2g 2
Eða; b; gÞ ¼ E 0 þ
where E0 is the ground-state energy at zero field, E0 ¼
_2 p2 _2 p2 _2 p2 þ þ . 2m L2 2m W 2 2m H 2
(6)
By minimizing E(a, b, g) with respect to a, b and g, the following equations of the parameters a, b and g are obtained: qLF sin y cos j ¼
m L2 ½1=ð2a2 Þ
1=ðp2
þ
a2 Þ
_2 a , þ 2a2 =ðp2 þ a2 Þ2 1=ð2 sinh2 aÞ
ð7Þ qWF sin y sin j ¼
_2 b , m W 2 ½1=ð2b2 Þ 1=ðp2 þ b2 Þ þ 2b2 =ðp2 þ b2 Þ2 1=ð2sinh2 bÞ
ð8Þ L2 oxo L2 ; W2 oyo W2 and H2 ozo H2 qHF cos y
elsewhere ð2Þ
using the infinite well model. Here, L, W and H are the length, width and height of the rectangular quantum box in the x-, y- and z-directions, respectively. When the direction of the electric field F~ in space is specified by its polar angle y and its azimuth angle j, the Hamiltonian is rewritten as p qxF sin y cos j qyF sin y sin j 2m qzF cos y þ V c ðx; y; zÞ.
¼
_2 g . m H ½1=ð2g2 Þ 1=ðp2 þ g2 Þ þ 2g2 =ðp2 þ g2 Þ2 1=ð2sinh2 gÞ 2
ð9Þ We find the Stark shift of the energy levels DE ¼ Eða; b; gÞ E 0 ¼
2
H¼
ð4Þ
where a, b and g are the variational parameters, N(a, b, g) is the normalization constant of the wave function, which is the function of a, b, g. A straightforward calculation yields the following expression for the expectation value of the Hamiltonian given by Eq. (3) using the variational wave function given by Eq (4)
2. Theory The calculation of the eigenstates of a quantum box in the presence of an electric field has an exactly solvable problem whose solutions involve the linear combinations of two independent Airy functions. However, these solutions are so complicated that it is difficult to analyze the asymptotic expansions of the Stark shift at low fields and high fields. In previous paper [20], we have given the transverse Stark effect in a rectangular quantum wire in the present of electric field by means of variational calculation method and effective-mass approximation, and found that the results given by the variational calculation method are almost in agreement with that given by the exact solution. Here, we continue to use the variational calculation method in our theoretical model. The Hamiltonian of a charged particle in a rectangular quantum box in the presence of an electric field is
787
ð3Þ
_2 a2 _2 b2 _2 g 2 þ þ 2 2 2m L 2m W 2m H 2 a 1 coth a þ qLF sin y cos j 2 þ a þ p2 2a 2
ARTICLE IN PRESS G. Wei et al. / Microelectronics Journal 39 (2008) 786–791
788
b 1 coth b þ qWF sin y sin j 2 þ 2 2b 2 b þp g 1 coth g þ qHF cos y 2 þ . g þ p2 2g 2
ð10Þ
By solving numerically Eqs. (7)–(10) for different sizes of the quantum box at various electric fields, we can study the Stark shift which is dependence of the sizes of the quantum box. It is worth noting that in the limits of low- or high-fields, the effect of the field on the Stark shift can be studied analytically. At low fields, parameters a, b and g are all small magnitudes according to Eqs. (7)–(9), so we can expand the denominator of the right side of Eqs. (7)–(9) in powers 1 of a, b and g, respectively. Noticing lim 2 sinh 2x1 2 16, the 2 x x!0
following expressions can be given after a straightforward calculation: qFm L3 sin y cos j 1 1 a¼ , (11) 6 p2 _2 qFm W 3 sin y sin j 1 1 b¼ , 6 p2 _2 qFm H 3 cos y 1 1 g¼ . 6 p2 _2
(12)
x!0
DE ¼
ð14Þ
By minimizing DE with respect to j and y, the following equations are obtained: sin2 y sin 2jðW 4 L4 Þ ¼ 0,
(15)
sin 2yðL4 cos2 j þ W 4 sin2 j H 4 Þ ¼ 0.
(16)
By solving Eqs (15) and (16), the following discussion results can be yielded: (i) for the cubic box (W=L=H), the maximum of the Stark shift DEmax is independent of the direction of the electric field; (ii) for the rectangular box, the DEmax is found when the direction of the electric field is along sides of the box. At high fields, parameters a, b and g are all large magnitudes according to Eqs. (7)–(9). Noticing 1 lim 2 sinh ! 0 and limx!1 coth x ! 1, similar to the 2 x
x!1
derived process in the low-field limit, we obtain 3m L2 qFL sin y cos j, 2_2
(18)
g3 ¼
3m H 2 qFH cos y, 2_2
(19)
qF ðL sin y cos j þ W sin y sin j þ H cos yÞ 2 5=3 2 2 2 1=3 3 qF _ þ 2 m
DE ¼
ðsin2=3 y cos2=3 j þ sin2=3 y sin2=3 j þ cos2=3 yÞ. ð20Þ 2/3
In Eq. (20), when the field is very high, FcF , the second term in the right side of Eq. (20) can be ignored, DE can be rewritten as DE ¼
qF ðL sin y cos j þ W sin y sin j þ H cos yÞ. 2 (21)
By minimizing DE with respect to j and y, the following equations are obtained: sin yðW cos j L sin jÞ ¼ 0,
Eqs. (11)–(13), we can obtain
a3 ¼
3m W 2 qFW sin y sin j, 2_2
(22)
(13)
Expanding the right side of Eq. (10) in powers of a, b, g and noticing lim coth x x1 þ x3 ; and substituting q2 F 2 m 4 2 ðL sin y cos2 j þ W 4 sin2 y sin2 j 2_2 1 1 2 4 2 þ H cos yÞ 2 . 6 p
b3 ¼
(17)
L cos j þ W sin j . (23) H Solving Eqs (22) and (23), the discussion results can be acquired. DEmax can be obtained when the electric field is directed along the diagonal line of either the rectangular box or cubic one. From Eqs. (14) and (21), we can see that the Stark shift is quadratic in the field at low fields and becomes almost linear in the field at high fields. These conclusions are the same as that obtained by the numerical solving in Ref. [18].
tan y ¼
3. Results and discussion In the following discussion, the energies are measured in effective electron Rydberg units Ry ¼ e2/2ka, the sizes of the quantum box are measured in effective electron Bohr radii a ¼ _2 k=m e2 and the electric field is measured in atomic units F0 ¼ e/ka2, where k is the dielectric constant of the semiconductor. In GaAs, the electron effective mass, m* ¼ 0.0665m0, the hole effective mass is mh ¼ 0:46m0 , where m0 is the fee electron mass and k ¼ 13.1. In Fig. 1, for various fields, DE is shown as a function of the angle y when the azimuth angle j ¼ p/4. Fig. 1(a) and (b) are for the cubic and rectangular boxes, respectively. From Fig. 1(a), we can see p that ffiffiffi DE always reaches a peak value when y ¼ arctanð 2Þ, namely the electric m * field F is along the direction of the diagonal line of the cubic box, this feature is independent of the field intensity. For the field directed along the height of the box, DE is the smallest. Take one closer look at Fig. 1(a) for the weak field F ¼ 10, the difference between the maximum of the Stark shift and its minimum is about 0.003Ry, very tiny.
ARTICLE IN PRESS G. Wei et al. / Microelectronics Journal 39 (2008) 786–791
Fig. 1. For j ¼ p/4, DE in a quantum box is shown as a function of the angle y for various electric field intensities and for given sizes of the boxes: (a) W ¼ L ¼ H ¼ 1a; (b) W ¼ L ¼ 1a, H ¼ 1.01a.
Fig. 2. For W ¼ L ¼ H ¼ 1a and W ¼ L ¼ 1a, H ¼ 1.2a, respectively, DE is shown as a function of the electric field directed along the diagonal of the box or along its height.
It is indicated that DE hardly depends upon the direction of the electric field for the very low fields as is mentioned before. From Fig. 1(b), however, one can see that the angle ym varies obviously with various field intensities for the rectangular box. For the low field directed along the height of the box, DE is the largest. Let the angle between the direction along the diagonal line of the rectangular box and the positive direction of z-axis is y0, with the field intensity the angle ym approaches toward the angle y0, that is, for the high field directed along the diagonal DE reaches the maximum. These features can also be seen in Fig. 2 which plots DE as a function of the electric field directed along the diagonal of the box or along its height. For the cubic
789
Fig. 3. For j ¼ p/4 and F ¼ 20F0, DE is shown as a function of the angle y for different height’s values of the square cross-sectional box of W ¼ L ¼ a.
box, at low fields, the Stark shift is almost independent of the two orientations of the electric field, the depicted curves overlap together. However, a completely different behavior takes place for high fields at which the Stark shift for the field directed along the diagonal is much larger than that for the field directed along the height. For the rectangular box, at low fields the Stark shift for the field directed along the height is larger; at high fields, the feature seen in the figure is same as that for the cubic box. In addition, we can see that the electric field is increasing function of DE. The Stark shift is quadratic in the electric field at low fields and becomes almost linear in the electric field at high fields. Below, let us take into account the effect of sizes of the rectangular quantum boxes on the largest Stark shift DEmax. Fig. 3, for different height’s values of the square cross-sectional box, show the Stark shift as a function of the angle y for azimuth angle j ¼ p/4 and F ¼ 20F0. By means of keeping the square cross-sectional sides fixed while expending the height’s value, we observe the dependence of sizes of the rectangular boxes on the largest Stark shift DEmax. From this figure, it is seen that DEmax is very sensitive to varying height of the box. When the height’s value equals the cross-sectional side’s value, DEmax appears for the direction of the field along the diagonal. Then the height’s value is expanded by a small magnitude of the height DH. DEmax can be observed, when the direction of field deviates the diagonal while moves toward the direction along the height (the positive direction of z-axis). The height’s value continues to increase to some value such as H ¼ 1.03a, DEmax is found when the field is directed along the height. Likewise, we can observe that DEmax increases with the size of the rectangular box. Additionally, for the electric field perpendicular to the z-axis of the boxes, the Stark shift of the energy is independent of the height H of the box; all curves in the figure focus on one point at y ¼ p/2.
ARTICLE IN PRESS G. Wei et al. / Microelectronics Journal 39 (2008) 786–791
790
Fig. 4. For various sizes of the cubic boxes, DEmax is shown as a function of the electric field.
The recombination rate is proportional to M 2eh which in Fig. 5 is plotted as a function of the electric field directed along the diagonal of the box or along its height. The particles are trapped in the cubic box for W ¼ L ¼ H ¼ 1a and in the rectangular box for W ¼ L ¼ 1a, H ¼ 1.5a, respectively. For the cubic box, M 2eh for the field directed along the diagonal decreases much faster than that for the field directed along the height, especially at high fields. For example, for F ¼ 100F0, M 2eh decreases by 26% for the field along the height and by 40% for the field along the diagonal. For the rectangular box, at low fields, the decrease of M 2eh is more for the direction of the field along the height of the box; at high fields the feature is identical to that for the cubic box. Comparing the result obtained for the cubic box with that for the rectangular, for the rectangular of H ¼ 1.5a, M 2eh decreases much faster. These features stem from the field-induced spatial separation of the electron and hole in the GaAs quantum box. The polarization effect in the box depends highly on the orientation of an external electric field. For the cubic box, the polarization effect is the most obvious for the field directed along the diagonal. However, for the rectangular box, the effect is the most pronounced for the low field directed along the height and for the high field directed along diagonal. 4. Conclusions
Fig. 5. For W ¼ L ¼ H ¼ 1a and W ¼ L ¼ 1a, H ¼ 1.5a, respectively, M 2eh is plotted as a function of the electric field directed along the diagonal of the box or along its height.
In Fig. 4, DEmax is shown as a function of the electric field for various sizes of the cubic boxes. From this figure, we can see that DEmax is increasing function of the electric field. The quantum sizes have high effect on the DEmax, especially for high fields. DEmax increases as the size of the box becomes large. In photoluminescence experiments, one is concerned with the recombination rate between the spatial confined electron and hole in an electric field, which is very significant. It is necessary that we calculate the overlap integral Meh defined as Z 1 M eh ¼ dx dy dz ce ðx; y; zÞch ðx; y; zÞ. 1
In the presence of the electric field, the Stark shift of electronic energies in a semiconducting rectangular quantum box has been investigated using variational calculation method and effective-mass approximation. The largest Stark shift in a quantum box can be found by means of changing the direction of an external electric field. For a cubic box, the Stark shift is largest when the field is directed along the diagonal. For a rectangular box, the largest Stark shift can be obtained when the field is applied along a side of box at low fields and applied along the diagonal at high fields. The quantum sizes have a demonstrable influence on the largest Stark shift. An obvious reduction of interband recombination is also yielded due to the large Stark shift of the electron and hole confined in a quantum box. Although to our knowledge, there are no available experimental data to compare with our theoretical results, we believe they provide an indication for design of some photoelectric devices constructed based on GaAs quantum box structures. References [1] G. Bastard, E.E. Mendez, L.L. Chang, L. Esaki, Phys. Rev. B 28 (1983) 3241. [2] J.A. Brum, G. Bastard, Phys. Rev. B 31 (1985) 3893. [3] D.A.B. Miller, D.S. Chemla, T.C. Damen, A.C. Gossard, W. Wiegmann, T.H. Wood, C.A. Burrus, Phys. Rev. B 32 (1985) 1043. [4] M. Matsuura, T. Kamizato, Phys. Rev. B 33 (1986) 8385. [5] Y.P. Feng, H.N. Spector, Phys. Rev. B 48 (1993) 1963. [6] Y.P. Feng, H.N. Spector, Phys. Status Solidi (b) 190 (1995) 211. [7] Y.P. Feng, H.S. Tan, H.N. Spector, Superlattices Microstruct. 17 (1995) 267.
ARTICLE IN PRESS G. Wei et al. / Microelectronics Journal 39 (2008) 786–791 [8] G.J. Vazques, M. del Castillo-Mussot, H.N. Spector, Phys. Status Solidi (b) 240 (2003) 561. [9] D. Huynh Thanh, D.B. Tran Thoai, Physica B 293 (2000) 1. [10] S. Benner, H. Haug, Phys. Rev. B 47 (1993) 15750. [11] R. Romestain, G. Fishman, Phys. Rev. B 49 (1994) 1774. [12] M. Pacheco, Z. Barticevic, Phys. Rev. B 55 (1997) 10688. [13] J. Lee, H.N. Spector, J. Appl. Phys. 97 (2005) 043511. [14] C.I. Mendoza, G.J. Vazquez, M.M. del Castillo-Mussot, H. Spector, Phys. Rev. B 71 (2005) 075330.
791
[15] T. Arakawa, Y. Kato, F. Sogawa, Y. Arakawa, Appl. Phys. Lett. 70 (1997) 646. [16] R. Rinaldi, R. Cingolani, L. DeCaro, M. Lomascolo, M. DiDio, L. Tapfer, U. Marti, F.K. Reinhart, J. Opt. Soc. Am. B 13 (1996) 1031. [17] H. Ham, H.N. Spector, Physica B 381 (2006) 53. [18] H.N. Spector, J. Lee, Physica B 393 (2007) 94. [19] S.S. Li, J.B. Xia, J. Appl. Phys. 101 (2007) 093716. [20] G.Y. Yi, G.Z. Wei, H.N. Wu, Phys. Status Solidi (b) 244 (2007) 4651.
Microelectronics Journal 39 (2008) 786–791 www.elsevier.com/locate/mejo
Stark effect of electrons in semiconducting rectangular quantum boxes Guozhu Weia,b,, Sheng Wanga, Guangyu Yia a
b
College of Sciences, Northeastern University, Shenyang 110004, China International Center for Material Physics, Academia Sinica, Shenyang 110015, China Received 27 September 2007; accepted 15 December 2007 Available online 31 January 2008
Abstract The Stark shift of the electronic energy levels in semiconducting rectangular quantum boxes with different sizes is investigated by the use of variational solutions to the effective-mass approximation for electric fields of various orientations with respect to the center axis of the box. The asymptotic expansions of the Stark shift are given in the limits of low and high fields, respectively; they clearly indicate that the Stark shift is a quadratic function of the electric field for low electric fields and is an approximate linear function of the electric field for high electric fields. Likewise, our results also show that the largest Stark shift is obtained for the field directed along the diagonal in a cubic box, and is found for the low field directed along a side of the box and for the high field along the diagonal in a rectangular one. The large Stark shift of the electron and hole trapped in a quantum box leads to an obvious reduction of the interband recombination. r 2008 Elsevier Ltd. All rights reserved. Keywords: Quantum boxes; Stark effect
1. Introduction In recent several decades, man-made low-dimensional solids yield challenges in microstructure materials science. It is important for fundamental physics and for the development of device concepts to excite electronic properties due to size quantization. The study of energy shifts of particles in semiconductor nanostructures of different geometries turns out to be a very important task since the shifts are associated with the spatial separation between electrons and holes in these structures which induces a reduction of interband recombination. It is useful to understand quantum lasers. Consequently, the quantum confined Stark effect (QCSE) of low-dimensional system has attracted considerable attention recently. There has been much work on Stark effect in quantum wells (QWs), quantum wires (QWRs), quantum dots (QDs) [1–19]. Ham and Spector [17] have investigated the effect of an electric field on the electronic energy levels in a semiconducting Corresponding author at: College of Sciences, Northeastern University, Shenyang 110004, China. E-mail address: [email protected] (G. Wei).
0026-2692/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.12.012
quantum disk by the use of variational solutions and found that the dependence of the Stark shift on the sizes of the disks is related to applied electric field directions with respect to disk axis. Spector and Lee [18] have calculated the effect of a transverse electric field on the ground and the first few excited states of the electrons confined in a cubical quantum box by using the expansion of the wave functions method and used these results to examine the effect of the electric field on the intersubband optical absorption in such quantum boxes. They found that the application of the electric field leads to a Stark shift of the electron energies, which is quadratic in the field at low fields but becomes almost linear in the field at high fields. There has been less work concerning the Stark effect in rectangular quantum boxes. Li and Xia [19] have calculated the Stark effect of the energy of a hydrogenic donor impurity in a rectangular parallelepiped-shaped quantum dot in the framework of effective-mass envelope-function theory using the plane wave basis. In this paper, we want to investigate the Stark shift of the energy levels in rectangular quantum boxes depends upon different sizes of the box for electric fields of various orientations with respect to the axis of the box by use of
ARTICLE IN PRESS G. Wei et al. / Microelectronics Journal 39 (2008) 786–791
variational calculation method, and emphasize how to obtain the largest Stark shift of the particle confined in a rectangular quantum box through changing the direction of the electric field. It is necessary to obtain the large Stark shift which leads to the high spatial separation of the electrons and holes inducing the reduction of interband recombination in these quantum structures. These results are relevant to any optical processes, such as emission and absorption. The paper is organized as follows. In Section 2, we describe the model for the rectangular quantum boxes and the variational method used to determine the Stark shift of the electric energies. In Section 3, the results are presented and discussed. In Section 4, conclusions are presented.
We take our variational wave function as px py pz cos cos Cðx; y; zÞ ¼ Nða; b; gÞ cos L W H ax gz by exp exp exp , L W H
* * ~ p2 * H¼ q F r þV c ð r Þ, (1) 2m ~ is the electric field, m* where ~ p is the carrier momentum, F and q are the carrier effective mass and charge, respec* tively, and r is the position vector of the electron. The confining potential is given by
V c ðx; y; zÞ ( 0 ¼ 1
_2 a2 _2 b2 _2 g 2 þ þ 2 2 2m L 2m W 2m H 2 a 1 coth a þ qLF sin y cos j 2 þ a þ p2 2a 2 b 1 coth b þ qWF sin y sin j 2 þ 2 b þ p2 2b g 1 coth g þ qHF cos y 2 þ , ð5Þ g þ p2 2g 2
Eða; b; gÞ ¼ E 0 þ
where E0 is the ground-state energy at zero field, E0 ¼
_2 p2 _2 p2 _2 p2 þ þ . 2m L2 2m W 2 2m H 2
(6)
By minimizing E(a, b, g) with respect to a, b and g, the following equations of the parameters a, b and g are obtained: qLF sin y cos j ¼
m L2 ½1=ð2a2 Þ
1=ðp2
þ
a2 Þ
_2 a , þ 2a2 =ðp2 þ a2 Þ2 1=ð2 sinh2 aÞ
ð7Þ qWF sin y sin j ¼
_2 b , m W 2 ½1=ð2b2 Þ 1=ðp2 þ b2 Þ þ 2b2 =ðp2 þ b2 Þ2 1=ð2sinh2 bÞ
ð8Þ L2 oxo L2 ; W2 oyo W2 and H2 ozo H2 qHF cos y
elsewhere ð2Þ
using the infinite well model. Here, L, W and H are the length, width and height of the rectangular quantum box in the x-, y- and z-directions, respectively. When the direction of the electric field F~ in space is specified by its polar angle y and its azimuth angle j, the Hamiltonian is rewritten as p qxF sin y cos j qyF sin y sin j 2m qzF cos y þ V c ðx; y; zÞ.
¼
_2 g . m H ½1=ð2g2 Þ 1=ðp2 þ g2 Þ þ 2g2 =ðp2 þ g2 Þ2 1=ð2sinh2 gÞ 2
ð9Þ We find the Stark shift of the energy levels DE ¼ Eða; b; gÞ E 0 ¼
2
H¼
ð4Þ
where a, b and g are the variational parameters, N(a, b, g) is the normalization constant of the wave function, which is the function of a, b, g. A straightforward calculation yields the following expression for the expectation value of the Hamiltonian given by Eq. (3) using the variational wave function given by Eq (4)
2. Theory The calculation of the eigenstates of a quantum box in the presence of an electric field has an exactly solvable problem whose solutions involve the linear combinations of two independent Airy functions. However, these solutions are so complicated that it is difficult to analyze the asymptotic expansions of the Stark shift at low fields and high fields. In previous paper [20], we have given the transverse Stark effect in a rectangular quantum wire in the present of electric field by means of variational calculation method and effective-mass approximation, and found that the results given by the variational calculation method are almost in agreement with that given by the exact solution. Here, we continue to use the variational calculation method in our theoretical model. The Hamiltonian of a charged particle in a rectangular quantum box in the presence of an electric field is
787
ð3Þ
_2 a2 _2 b2 _2 g 2 þ þ 2 2 2m L 2m W 2m H 2 a 1 coth a þ qLF sin y cos j 2 þ a þ p2 2a 2
ARTICLE IN PRESS G. Wei et al. / Microelectronics Journal 39 (2008) 786–791
788
b 1 coth b þ qWF sin y sin j 2 þ 2 2b 2 b þp g 1 coth g þ qHF cos y 2 þ . g þ p2 2g 2
ð10Þ
By solving numerically Eqs. (7)–(10) for different sizes of the quantum box at various electric fields, we can study the Stark shift which is dependence of the sizes of the quantum box. It is worth noting that in the limits of low- or high-fields, the effect of the field on the Stark shift can be studied analytically. At low fields, parameters a, b and g are all small magnitudes according to Eqs. (7)–(9), so we can expand the denominator of the right side of Eqs. (7)–(9) in powers 1 of a, b and g, respectively. Noticing lim 2 sinh 2x1 2 16, the 2 x x!0
following expressions can be given after a straightforward calculation: qFm L3 sin y cos j 1 1 a¼ , (11) 6 p2 _2 qFm W 3 sin y sin j 1 1 b¼ , 6 p2 _2 qFm H 3 cos y 1 1 g¼ . 6 p2 _2
(12)
x!0
DE ¼
ð14Þ
By minimizing DE with respect to j and y, the following equations are obtained: sin2 y sin 2jðW 4 L4 Þ ¼ 0,
(15)
sin 2yðL4 cos2 j þ W 4 sin2 j H 4 Þ ¼ 0.
(16)
By solving Eqs (15) and (16), the following discussion results can be yielded: (i) for the cubic box (W=L=H), the maximum of the Stark shift DEmax is independent of the direction of the electric field; (ii) for the rectangular box, the DEmax is found when the direction of the electric field is along sides of the box. At high fields, parameters a, b and g are all large magnitudes according to Eqs. (7)–(9). Noticing 1 lim 2 sinh ! 0 and limx!1 coth x ! 1, similar to the 2 x
x!1
derived process in the low-field limit, we obtain 3m L2 qFL sin y cos j, 2_2
(18)
g3 ¼
3m H 2 qFH cos y, 2_2
(19)
qF ðL sin y cos j þ W sin y sin j þ H cos yÞ 2 5=3 2 2 2 1=3 3 qF _ þ 2 m
DE ¼
ðsin2=3 y cos2=3 j þ sin2=3 y sin2=3 j þ cos2=3 yÞ. ð20Þ 2/3
In Eq. (20), when the field is very high, FcF , the second term in the right side of Eq. (20) can be ignored, DE can be rewritten as DE ¼
qF ðL sin y cos j þ W sin y sin j þ H cos yÞ. 2 (21)
By minimizing DE with respect to j and y, the following equations are obtained: sin yðW cos j L sin jÞ ¼ 0,
Eqs. (11)–(13), we can obtain
a3 ¼
3m W 2 qFW sin y sin j, 2_2
(22)
(13)
Expanding the right side of Eq. (10) in powers of a, b, g and noticing lim coth x x1 þ x3 ; and substituting q2 F 2 m 4 2 ðL sin y cos2 j þ W 4 sin2 y sin2 j 2_2 1 1 2 4 2 þ H cos yÞ 2 . 6 p
b3 ¼
(17)
L cos j þ W sin j . (23) H Solving Eqs (22) and (23), the discussion results can be acquired. DEmax can be obtained when the electric field is directed along the diagonal line of either the rectangular box or cubic one. From Eqs. (14) and (21), we can see that the Stark shift is quadratic in the field at low fields and becomes almost linear in the field at high fields. These conclusions are the same as that obtained by the numerical solving in Ref. [18].
tan y ¼
3. Results and discussion In the following discussion, the energies are measured in effective electron Rydberg units Ry ¼ e2/2ka, the sizes of the quantum box are measured in effective electron Bohr radii a ¼ _2 k=m e2 and the electric field is measured in atomic units F0 ¼ e/ka2, where k is the dielectric constant of the semiconductor. In GaAs, the electron effective mass, m* ¼ 0.0665m0, the hole effective mass is mh ¼ 0:46m0 , where m0 is the fee electron mass and k ¼ 13.1. In Fig. 1, for various fields, DE is shown as a function of the angle y when the azimuth angle j ¼ p/4. Fig. 1(a) and (b) are for the cubic and rectangular boxes, respectively. From Fig. 1(a), we can see p that ffiffiffi DE always reaches a peak value when y ¼ arctanð 2Þ, namely the electric m * field F is along the direction of the diagonal line of the cubic box, this feature is independent of the field intensity. For the field directed along the height of the box, DE is the smallest. Take one closer look at Fig. 1(a) for the weak field F ¼ 10, the difference between the maximum of the Stark shift and its minimum is about 0.003Ry, very tiny.
ARTICLE IN PRESS G. Wei et al. / Microelectronics Journal 39 (2008) 786–791
Fig. 1. For j ¼ p/4, DE in a quantum box is shown as a function of the angle y for various electric field intensities and for given sizes of the boxes: (a) W ¼ L ¼ H ¼ 1a; (b) W ¼ L ¼ 1a, H ¼ 1.01a.
Fig. 2. For W ¼ L ¼ H ¼ 1a and W ¼ L ¼ 1a, H ¼ 1.2a, respectively, DE is shown as a function of the electric field directed along the diagonal of the box or along its height.
It is indicated that DE hardly depends upon the direction of the electric field for the very low fields as is mentioned before. From Fig. 1(b), however, one can see that the angle ym varies obviously with various field intensities for the rectangular box. For the low field directed along the height of the box, DE is the largest. Let the angle between the direction along the diagonal line of the rectangular box and the positive direction of z-axis is y0, with the field intensity the angle ym approaches toward the angle y0, that is, for the high field directed along the diagonal DE reaches the maximum. These features can also be seen in Fig. 2 which plots DE as a function of the electric field directed along the diagonal of the box or along its height. For the cubic
789
Fig. 3. For j ¼ p/4 and F ¼ 20F0, DE is shown as a function of the angle y for different height’s values of the square cross-sectional box of W ¼ L ¼ a.
box, at low fields, the Stark shift is almost independent of the two orientations of the electric field, the depicted curves overlap together. However, a completely different behavior takes place for high fields at which the Stark shift for the field directed along the diagonal is much larger than that for the field directed along the height. For the rectangular box, at low fields the Stark shift for the field directed along the height is larger; at high fields, the feature seen in the figure is same as that for the cubic box. In addition, we can see that the electric field is increasing function of DE. The Stark shift is quadratic in the electric field at low fields and becomes almost linear in the electric field at high fields. Below, let us take into account the effect of sizes of the rectangular quantum boxes on the largest Stark shift DEmax. Fig. 3, for different height’s values of the square cross-sectional box, show the Stark shift as a function of the angle y for azimuth angle j ¼ p/4 and F ¼ 20F0. By means of keeping the square cross-sectional sides fixed while expending the height’s value, we observe the dependence of sizes of the rectangular boxes on the largest Stark shift DEmax. From this figure, it is seen that DEmax is very sensitive to varying height of the box. When the height’s value equals the cross-sectional side’s value, DEmax appears for the direction of the field along the diagonal. Then the height’s value is expanded by a small magnitude of the height DH. DEmax can be observed, when the direction of field deviates the diagonal while moves toward the direction along the height (the positive direction of z-axis). The height’s value continues to increase to some value such as H ¼ 1.03a, DEmax is found when the field is directed along the height. Likewise, we can observe that DEmax increases with the size of the rectangular box. Additionally, for the electric field perpendicular to the z-axis of the boxes, the Stark shift of the energy is independent of the height H of the box; all curves in the figure focus on one point at y ¼ p/2.
ARTICLE IN PRESS G. Wei et al. / Microelectronics Journal 39 (2008) 786–791
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Fig. 4. For various sizes of the cubic boxes, DEmax is shown as a function of the electric field.
The recombination rate is proportional to M 2eh which in Fig. 5 is plotted as a function of the electric field directed along the diagonal of the box or along its height. The particles are trapped in the cubic box for W ¼ L ¼ H ¼ 1a and in the rectangular box for W ¼ L ¼ 1a, H ¼ 1.5a, respectively. For the cubic box, M 2eh for the field directed along the diagonal decreases much faster than that for the field directed along the height, especially at high fields. For example, for F ¼ 100F0, M 2eh decreases by 26% for the field along the height and by 40% for the field along the diagonal. For the rectangular box, at low fields, the decrease of M 2eh is more for the direction of the field along the height of the box; at high fields the feature is identical to that for the cubic box. Comparing the result obtained for the cubic box with that for the rectangular, for the rectangular of H ¼ 1.5a, M 2eh decreases much faster. These features stem from the field-induced spatial separation of the electron and hole in the GaAs quantum box. The polarization effect in the box depends highly on the orientation of an external electric field. For the cubic box, the polarization effect is the most obvious for the field directed along the diagonal. However, for the rectangular box, the effect is the most pronounced for the low field directed along the height and for the high field directed along diagonal. 4. Conclusions
Fig. 5. For W ¼ L ¼ H ¼ 1a and W ¼ L ¼ 1a, H ¼ 1.5a, respectively, M 2eh is plotted as a function of the electric field directed along the diagonal of the box or along its height.
In Fig. 4, DEmax is shown as a function of the electric field for various sizes of the cubic boxes. From this figure, we can see that DEmax is increasing function of the electric field. The quantum sizes have high effect on the DEmax, especially for high fields. DEmax increases as the size of the box becomes large. In photoluminescence experiments, one is concerned with the recombination rate between the spatial confined electron and hole in an electric field, which is very significant. It is necessary that we calculate the overlap integral Meh defined as Z 1 M eh ¼ dx dy dz ce ðx; y; zÞch ðx; y; zÞ. 1
In the presence of the electric field, the Stark shift of electronic energies in a semiconducting rectangular quantum box has been investigated using variational calculation method and effective-mass approximation. The largest Stark shift in a quantum box can be found by means of changing the direction of an external electric field. For a cubic box, the Stark shift is largest when the field is directed along the diagonal. For a rectangular box, the largest Stark shift can be obtained when the field is applied along a side of box at low fields and applied along the diagonal at high fields. The quantum sizes have a demonstrable influence on the largest Stark shift. An obvious reduction of interband recombination is also yielded due to the large Stark shift of the electron and hole confined in a quantum box. Although to our knowledge, there are no available experimental data to compare with our theoretical results, we believe they provide an indication for design of some photoelectric devices constructed based on GaAs quantum box structures. References [1] G. Bastard, E.E. Mendez, L.L. Chang, L. Esaki, Phys. Rev. B 28 (1983) 3241. [2] J.A. Brum, G. Bastard, Phys. Rev. B 31 (1985) 3893. [3] D.A.B. Miller, D.S. Chemla, T.C. Damen, A.C. Gossard, W. Wiegmann, T.H. Wood, C.A. Burrus, Phys. Rev. B 32 (1985) 1043. [4] M. Matsuura, T. Kamizato, Phys. Rev. B 33 (1986) 8385. [5] Y.P. Feng, H.N. Spector, Phys. Rev. B 48 (1993) 1963. [6] Y.P. Feng, H.N. Spector, Phys. Status Solidi (b) 190 (1995) 211. [7] Y.P. Feng, H.S. Tan, H.N. Spector, Superlattices Microstruct. 17 (1995) 267.
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