Transverse Stark effect in the optical absorption in a square semiconducting quantum wire

Sheng Wang; Yun Kang; Chunjie Han

doi:10.1088/1674-4926/34/10/102001

J. Semicond. > 2013, Volume 34 > Issue 10 > 102001, doi: 10.1088/1674-4926/34/10/102001

SEMICONDUCTOR PHYSICS

Transverse Stark effect in the optical absorption in a square semiconducting quantum wire

Sheng Wang, Yun Kang and Chunjie Han

+ Author Affiliations

Corresponding author: Wang Sheng, wangsheng10@126.com

Abstract: The ground and few excited states of the electrons confined in a square GaAs quantum wire (QW) subjected to an external transverse electric field are investigated using the finite difference method within the effective-mass approximation. When the transverse electric field is applied along a side of the square quantum wire, the calculation of the eigenstates of the quantum wire has an exactly solvable problem whose solutions involve the linear combinations of two independent Airy functions. Compared with the exact analytical results using Airy functions, the results obtained by the use of the finite difference method in terms of the eigenstates of the particle in the QW are in excellent agreement. Subsequently, it is considered that the eigenstates of the particle depend on the orientations of the electric field with respect to the center axis of the QW. It is interesting that the peak value of the energy is found for the field directed along the diagonal in the QW, which can lead to a large energy shift. Meanwhile, dependence of the optical absorption phenomenon in the square QW on the optical field and applied electric field is investigated. It is shown that the optical absorption spectrum depends highly upon the polarization of the optical field and the applied electric field intensities and orientations.

Key words: finite difference method, quantum wire, optical absorption, Stark effect

References

[1]	Feng Y P, Spector H N. Exciton energies as a function of electric field:confined quantum Stark effect. Phys Rev B, 1993, 48:1963 doi: 10.1103/PhysRevB.48.1963
[2]	Jiao Z K, Chen Q H, Ren Y H. An effective approach for two-dimensional polarons in an asymmetric quantum dot. Chin Phys B, 1998, 7:598 http://cpb.iphy.ac.cn/EN/Y1998/V7/I8/598
[3]	Liu J J, Chen X F, Li S S. Binding energy of biexcitons in GaAs quantum-well wires. Chin Phys Lett, 2004, 21:2259 doi: 10.1088/0256-307X/21/11/051
[4]	Spector H N, Lee J. Stark effect in the optical absorption in cubical quantum boxes. Physica B, 2007, 94:393 http://www.sciencedirect.com/science/article/pii/S0921452606019533
[5]	Liu J J, Di B, Yang G C, et al. Variational calculations of neutral bound excitons in GaAs quantum-well wires. Chin Phys Lett, 2004, 21:919 doi: 10.1088/0256-307X/21/5/044
[6]	Wu Y F, Baja K K, Liang X X. On the binding energies of excitons in polar quantum well structures in a weak electric field. Chin Phys B, 2005, 14:2314 doi: 10.1088/1009-1963/14/11/028
[7]	Wang S, Wei G Z, Yi G Y. Binding energies of hydrogenic impurities on-center and off-center in cylindrical quantum dots under electric and magnetic fields. Int J Mod Phys B, 2010, 24:4293 doi: 10.1142/S0217979210055937
[8]	Zhang Y T, Li Y C, Li K H. Experimental investigation of the velocity effect on adhesion forces with an atomic force microscope. Chin Phys Lett, 2004, 21:616 doi: 10.1088/0256-307X/21/4/007
[9]	Shi H Y, Guo R H, Sun X D. Theoretical study of quantum confined Stark shift in InAs/GaAs quantum dots. Chin Phys B, 2004, 13:2141 doi: 10.1088/1009-1963/13/12/029
[10]	Wang S, Kang Y, Han Y. Transverse Stark effect of electrons in GaAs semiconducting quantum boxes. J At Mol Sci, 2011, 2:162 http://www.global-sci.org/jams/volumes/v2n2/pdf/022-162.pdf
[11]	Sheng W D, Leburton J P. Absence of correlation between built-in electric dipole moment and quantum Stark effect in single InAs/GaAs self-assembled quantum dots. Phys Rev B, 2003, 67:125308 doi: 10.1103/PhysRevB.67.125308
[12]	Zhang T Y, Zhao W, Zhu H Y, et al. A full numerical calculation of the Franz-Keldysh effect on magnetoexcitons in a bulk semiconductor. Chin Phys B, 2006, 15:2151 doi: 10.1088/1009-1963/15/9/041
[13]	Pokutnyi S I, Jacak L, Misiewicz J, et al. Stark effect in semiconductor quantum dots. J Appl Phys, 2004, 96:1115 doi: 10.1063/1.1759791
[14]	Cao X L, Li Z Y, Yao J Q, et al. Intersubband absorption with difference-frequency generation in GaAs asymmetric quantum wells. Chin Phys B, 2012, 21:084207 doi: 10.1088/1674-1056/21/8/084207
[15]	Vázquez G J, Castillo M M, Spector H N. Transverse Stark effect of electrons in a semiconducting quantum wire. Phys Status Solidi B, 2003, 240:561 doi: 10.1002/(ISSN)1521-3951
[16]	Vázquez G J, Castillo M M, Montemayor A J, et al. Transverse Stark effect of electrons in a hollow semiconducting quantum wire. Physica E, 2006, 33:240 doi: 10.1016/j.physe.2006.02.036
[17]	Dupertuis M A, Martinet E, Weman H, et al. Quantum confined Stark effect in quantum wires:wave function splitting and cascading. Europhys Lett, 1998, 44:759 doi: 10.1209/epl/i1998-00538-7
[18]	Huynh T D, Tran T D B. Influence of external electric field on the band-gap renormalization and excitonic absorption spectra of highly excited quantum-well wires. Physica B, 2000, 293:1 doi: 10.1016/S0921-4526(00)00544-5
[19]	Benner S, Haug H. Influence of external electric and magnetic fields on the excitonic absorption spectra of quantum-well wires. Phys Rev B, 1993, 47:15750 doi: 10.1103/PhysRevB.47.15750
[20]	Arakawa T, Kato Y, Sogawa F, et al. Photoluminescence studies of GaAs quantum wires with quantum confined Stark effect. Appl Phys Lett, 1997, 70:646 doi: 10.1063/1.118295
[21]	Rinaldi R, Cingolani R, De C L, et al. Optical spectroscopy of InGaAs/GaAs V-shaped quantum wires. J Opt Soc Am B, 1996, 13:1031 doi: 10.1364/JOSAB.13.001031
[22]	Lee J, Spector H N. Stark effect in the optical absorption in quantum wires. J Appl Phys, 2005, 97:043511 doi: 10.1063/1.1849430
[23]	Xia C X, Wei S Y, Spector H N. Franz-Keldysh effect in ZnO quantum wire. Physica E, 2010, 42:2065 doi: 10.1016/j.physe.2010.03.027
[24]	Yi G Y, Wei G Z, Wu H N. Transverse Stark effect in a rectangular semiconducting quantum wire. Phys Status Solidi B, 2007, 224:4651 doi: 10.1134/S1063739710060065
[25]	Zhang H, Wang X, Zhao J F, et al. The binding energy of a hydrogenic impurity in self-assembled double quantum dots. Chin Phys B, 2011, 20:127301 doi: 10.1088/1674-1056/20/12/127301
[26]	Riva C, Escorcia R A, Govorov A O, et al. Charged donors in quantum dots:finite difference and fractional dimensions results. Phys Rev B, 2004, 69:245306 doi: 10.1103/PhysRevB.69.245306

Fig. 1. The energy levels of electrons in the quantum wire are shown as a function of the electric field for the polar angle $\theta =0$ . The ground state $j$ $=$ 1 and the excited states ( $j$ $>$ 2) are obtained respectively by use of the exact solutions in terms of Airy functions (circles) and finite difference numerical method (solid lines).

DownLoad: Full-Size Img PowerPoint

Fig. 2. The electron probability density of the ground state $j=$ 1 and excited states (j > 2) as functions of x position for the electric fields $F=8F_0$, $\theta =0$ respectively by use of the exact solutions in terms of Airy functions and finite difference numerical method

DownLoad: Full-Size Img PowerPoint

Fig. 3. Several energy levels ( $j$ $=$ 1, 2, 3, 4, 5) as functions of the orientations of the electric field for the electric field intensity $F=8F_0$ .

DownLoad: Full-Size Img PowerPoint

Fig. 4. The electron probability density of the excited state j = 4 as functions of the plane x-y positions for the different electric fields

DownLoad: Full-Size Img PowerPoint

Fig. 5. The optical absorption as a function of the photon energy (a) for three different polarizations (polarizations- $x$ $\boldsymbol{\varepsilon}=(1, 0)$ , polarizations- $y$ $\boldsymbol{\varepsilon} =(0, 1)$ , polarizations- $xy$ $\boldsymbol{\varepsilon} =\frac{1}{\sqrt 2}(1, 1))$ with a fixed electric field of $F=8F_0$ , $\theta =0$ ; (b) for three different polarizations with a fixed electric field of $F=8F_0$ , $\theta =\frac{\pi }{4}$ ; (c) for the $x$ - $y$ polarization ( $\boldsymbol{\varepsilon}=\frac{1}{\sqrt 2 }(1, 1))$ with different electric fields $F=0, F=8F_0$ along the $x$ direction.

DownLoad: Full-Size Img PowerPoint

[1]	Feng Y P, Spector H N. Exciton energies as a function of electric field:confined quantum Stark effect. Phys Rev B, 1993, 48:1963 doi: 10.1103/PhysRevB.48.1963
[2]	Jiao Z K, Chen Q H, Ren Y H. An effective approach for two-dimensional polarons in an asymmetric quantum dot. Chin Phys B, 1998, 7:598 http://cpb.iphy.ac.cn/EN/Y1998/V7/I8/598
[3]	Liu J J, Chen X F, Li S S. Binding energy of biexcitons in GaAs quantum-well wires. Chin Phys Lett, 2004, 21:2259 doi: 10.1088/0256-307X/21/11/051
[4]	Spector H N, Lee J. Stark effect in the optical absorption in cubical quantum boxes. Physica B, 2007, 94:393 http://www.sciencedirect.com/science/article/pii/S0921452606019533
[5]	Liu J J, Di B, Yang G C, et al. Variational calculations of neutral bound excitons in GaAs quantum-well wires. Chin Phys Lett, 2004, 21:919 doi: 10.1088/0256-307X/21/5/044
[6]	Wu Y F, Baja K K, Liang X X. On the binding energies of excitons in polar quantum well structures in a weak electric field. Chin Phys B, 2005, 14:2314 doi: 10.1088/1009-1963/14/11/028
[7]	Wang S, Wei G Z, Yi G Y. Binding energies of hydrogenic impurities on-center and off-center in cylindrical quantum dots under electric and magnetic fields. Int J Mod Phys B, 2010, 24:4293 doi: 10.1142/S0217979210055937
[8]	Zhang Y T, Li Y C, Li K H. Experimental investigation of the velocity effect on adhesion forces with an atomic force microscope. Chin Phys Lett, 2004, 21:616 doi: 10.1088/0256-307X/21/4/007
[9]	Shi H Y, Guo R H, Sun X D. Theoretical study of quantum confined Stark shift in InAs/GaAs quantum dots. Chin Phys B, 2004, 13:2141 doi: 10.1088/1009-1963/13/12/029
[10]	Wang S, Kang Y, Han Y. Transverse Stark effect of electrons in GaAs semiconducting quantum boxes. J At Mol Sci, 2011, 2:162 http://www.global-sci.org/jams/volumes/v2n2/pdf/022-162.pdf
[11]	Sheng W D, Leburton J P. Absence of correlation between built-in electric dipole moment and quantum Stark effect in single InAs/GaAs self-assembled quantum dots. Phys Rev B, 2003, 67:125308 doi: 10.1103/PhysRevB.67.125308
[12]	Zhang T Y, Zhao W, Zhu H Y, et al. A full numerical calculation of the Franz-Keldysh effect on magnetoexcitons in a bulk semiconductor. Chin Phys B, 2006, 15:2151 doi: 10.1088/1009-1963/15/9/041
[13]	Pokutnyi S I, Jacak L, Misiewicz J, et al. Stark effect in semiconductor quantum dots. J Appl Phys, 2004, 96:1115 doi: 10.1063/1.1759791
[14]	Cao X L, Li Z Y, Yao J Q, et al. Intersubband absorption with difference-frequency generation in GaAs asymmetric quantum wells. Chin Phys B, 2012, 21:084207 doi: 10.1088/1674-1056/21/8/084207
[15]	Vázquez G J, Castillo M M, Spector H N. Transverse Stark effect of electrons in a semiconducting quantum wire. Phys Status Solidi B, 2003, 240:561 doi: 10.1002/(ISSN)1521-3951
[16]	Vázquez G J, Castillo M M, Montemayor A J, et al. Transverse Stark effect of electrons in a hollow semiconducting quantum wire. Physica E, 2006, 33:240 doi: 10.1016/j.physe.2006.02.036
[17]	Dupertuis M A, Martinet E, Weman H, et al. Quantum confined Stark effect in quantum wires:wave function splitting and cascading. Europhys Lett, 1998, 44:759 doi: 10.1209/epl/i1998-00538-7
[18]	Huynh T D, Tran T D B. Influence of external electric field on the band-gap renormalization and excitonic absorption spectra of highly excited quantum-well wires. Physica B, 2000, 293:1 doi: 10.1016/S0921-4526(00)00544-5
[19]	Benner S, Haug H. Influence of external electric and magnetic fields on the excitonic absorption spectra of quantum-well wires. Phys Rev B, 1993, 47:15750 doi: 10.1103/PhysRevB.47.15750
[20]	Arakawa T, Kato Y, Sogawa F, et al. Photoluminescence studies of GaAs quantum wires with quantum confined Stark effect. Appl Phys Lett, 1997, 70:646 doi: 10.1063/1.118295
[21]	Rinaldi R, Cingolani R, De C L, et al. Optical spectroscopy of InGaAs/GaAs V-shaped quantum wires. J Opt Soc Am B, 1996, 13:1031 doi: 10.1364/JOSAB.13.001031
[22]	Lee J, Spector H N. Stark effect in the optical absorption in quantum wires. J Appl Phys, 2005, 97:043511 doi: 10.1063/1.1849430
[23]	Xia C X, Wei S Y, Spector H N. Franz-Keldysh effect in ZnO quantum wire. Physica E, 2010, 42:2065 doi: 10.1016/j.physe.2010.03.027
[24]	Yi G Y, Wei G Z, Wu H N. Transverse Stark effect in a rectangular semiconducting quantum wire. Phys Status Solidi B, 2007, 224:4651 doi: 10.1134/S1063739710060065
[25]	Zhang H, Wang X, Zhao J F, et al. The binding energy of a hydrogenic impurity in self-assembled double quantum dots. Chin Phys B, 2011, 20:127301 doi: 10.1088/1674-1056/20/12/127301
[26]	Riva C, Escorcia R A, Govorov A O, et al. Charged donors in quantum dots:finite difference and fractional dimensions results. Phys Rev B, 2004, 69:245306 doi: 10.1103/PhysRevB.69.245306

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Received: 10 March 2013 Revised: 28 April 2013 Online: Published: 01 October 2013

Article Navigation > Journal of Semiconductors > 2013 > 34(10): 102001

Sheng Wang, Yun Kang, Chunjie Han. Transverse Stark effect in the optical absorption in a square semiconducting quantum wire[J]. Journal of Semiconductors, 2013, 34(10): 102001. doi: 10.1088/1674-4926/34/10/102001 S Wang, Y Kang, C J Han. Transverse Stark effect in the optical absorption in a square semiconducting quantum wire[J]. J. Semicond., 2013, 34(10): 102001. doi: 10.1088/1674-4926/34/10/102001.Export: BibTex EndNote

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Sheng Wang, Yun Kang, Chunjie Han. Transverse Stark effect in the optical absorption in a square semiconducting quantum wire[J]. Journal of Semiconductors, 2013, 34(10): 102001. doi: 10.1088/1674-4926/34/10/102001

S Wang, Y Kang, C J Han. Transverse Stark effect in the optical absorption in a square semiconducting quantum wire[J]. J. Semicond., 2013, 34(10): 102001. doi: 10.1088/1674-4926/34/10/102001.

Export: BibTex EndNote

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S Wang, Y Kang, C J Han. Transverse Stark effect in the optical absorption in a square semiconducting quantum wire[J]. J. Semicond., 2013, 34(10): 102001. doi: 10.1088/1674-4926/34/10/102001.

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Transverse Stark effect in the optical absorption in a square semiconducting quantum wire

doi: 10.1088/1674-4926/34/10/102001

College of Electronic Science, Northeastern Petroleum University, Daqing 163318, China

Funds:

the Northeastern Petroleum University Youth Science Foundation of China 2012QN128

Project supported by the National Natural Science Foundation of China (No. 41202146), the Natural Science Foundation of Heilongjiang Province of China (No. F201211), and the Northeastern Petroleum University Youth Science Foundation of China (No. 2012QN128)

the National Natural Science Foundation of China 41202146

the Natural Science Foundation of Heilongjiang Province of China F201211

More Information

Corresponding author: Wang Sheng, wangsheng10@126.com
Received Date: 2013-03-10
Revised Date: 2013-04-28
Published Date: 2013-10-01

Abstract

Abstract

The ground and few excited states of the electrons confined in a square GaAs quantum wire (QW) subjected to an external transverse electric field are investigated using the finite difference method within the effective-mass approximation. When the transverse electric field is applied along a side of the square quantum wire, the calculation of the eigenstates of the quantum wire has an exactly solvable problem whose solutions involve the linear combinations of two independent Airy functions. Compared with the exact analytical results using Airy functions, the results obtained by the use of the finite difference method in terms of the eigenstates of the particle in the QW are in excellent agreement. Subsequently, it is considered that the eigenstates of the particle depend on the orientations of the electric field with respect to the center axis of the QW. It is interesting that the peak value of the energy is found for the field directed along the diagonal in the QW, which can lead to a large energy shift. Meanwhile, dependence of the optical absorption phenomenon in the square QW on the optical field and applied electric field is investigated. It is shown that the optical absorption spectrum depends highly upon the polarization of the optical field and the applied electric field intensities and orientations.
- finite difference method,
- quantum wire,
- optical absorption,
- Stark effect

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References(26)

References

[1]	Feng Y P, Spector H N. Exciton energies as a function of electric field:confined quantum Stark effect. Phys Rev B, 1993, 48:1963 doi: 10.1103/PhysRevB.48.1963
[2]	Jiao Z K, Chen Q H, Ren Y H. An effective approach for two-dimensional polarons in an asymmetric quantum dot. Chin Phys B, 1998, 7:598 http://cpb.iphy.ac.cn/EN/Y1998/V7/I8/598
[3]	Liu J J, Chen X F, Li S S. Binding energy of biexcitons in GaAs quantum-well wires. Chin Phys Lett, 2004, 21:2259 doi: 10.1088/0256-307X/21/11/051
[4]	Spector H N, Lee J. Stark effect in the optical absorption in cubical quantum boxes. Physica B, 2007, 94:393 http://www.sciencedirect.com/science/article/pii/S0921452606019533
[5]	Liu J J, Di B, Yang G C, et al. Variational calculations of neutral bound excitons in GaAs quantum-well wires. Chin Phys Lett, 2004, 21:919 doi: 10.1088/0256-307X/21/5/044
[6]	Wu Y F, Baja K K, Liang X X. On the binding energies of excitons in polar quantum well structures in a weak electric field. Chin Phys B, 2005, 14:2314 doi: 10.1088/1009-1963/14/11/028
[7]	Wang S, Wei G Z, Yi G Y. Binding energies of hydrogenic impurities on-center and off-center in cylindrical quantum dots under electric and magnetic fields. Int J Mod Phys B, 2010, 24:4293 doi: 10.1142/S0217979210055937
[8]	Zhang Y T, Li Y C, Li K H. Experimental investigation of the velocity effect on adhesion forces with an atomic force microscope. Chin Phys Lett, 2004, 21:616 doi: 10.1088/0256-307X/21/4/007
[9]	Shi H Y, Guo R H, Sun X D. Theoretical study of quantum confined Stark shift in InAs/GaAs quantum dots. Chin Phys B, 2004, 13:2141 doi: 10.1088/1009-1963/13/12/029
[10]	Wang S, Kang Y, Han Y. Transverse Stark effect of electrons in GaAs semiconducting quantum boxes. J At Mol Sci, 2011, 2:162 http://www.global-sci.org/jams/volumes/v2n2/pdf/022-162.pdf
[11]	Sheng W D, Leburton J P. Absence of correlation between built-in electric dipole moment and quantum Stark effect in single InAs/GaAs self-assembled quantum dots. Phys Rev B, 2003, 67:125308 doi: 10.1103/PhysRevB.67.125308
[12]	Zhang T Y, Zhao W, Zhu H Y, et al. A full numerical calculation of the Franz-Keldysh effect on magnetoexcitons in a bulk semiconductor. Chin Phys B, 2006, 15:2151 doi: 10.1088/1009-1963/15/9/041
[13]	Pokutnyi S I, Jacak L, Misiewicz J, et al. Stark effect in semiconductor quantum dots. J Appl Phys, 2004, 96:1115 doi: 10.1063/1.1759791
[14]	Cao X L, Li Z Y, Yao J Q, et al. Intersubband absorption with difference-frequency generation in GaAs asymmetric quantum wells. Chin Phys B, 2012, 21:084207 doi: 10.1088/1674-1056/21/8/084207
[15]	Vázquez G J, Castillo M M, Spector H N. Transverse Stark effect of electrons in a semiconducting quantum wire. Phys Status Solidi B, 2003, 240:561 doi: 10.1002/(ISSN)1521-3951
[16]	Vázquez G J, Castillo M M, Montemayor A J, et al. Transverse Stark effect of electrons in a hollow semiconducting quantum wire. Physica E, 2006, 33:240 doi: 10.1016/j.physe.2006.02.036
[17]	Dupertuis M A, Martinet E, Weman H, et al. Quantum confined Stark effect in quantum wires:wave function splitting and cascading. Europhys Lett, 1998, 44:759 doi: 10.1209/epl/i1998-00538-7
[18]	Huynh T D, Tran T D B. Influence of external electric field on the band-gap renormalization and excitonic absorption spectra of highly excited quantum-well wires. Physica B, 2000, 293:1 doi: 10.1016/S0921-4526(00)00544-5
[19]	Benner S, Haug H. Influence of external electric and magnetic fields on the excitonic absorption spectra of quantum-well wires. Phys Rev B, 1993, 47:15750 doi: 10.1103/PhysRevB.47.15750
[20]	Arakawa T, Kato Y, Sogawa F, et al. Photoluminescence studies of GaAs quantum wires with quantum confined Stark effect. Appl Phys Lett, 1997, 70:646 doi: 10.1063/1.118295
[21]	Rinaldi R, Cingolani R, De C L, et al. Optical spectroscopy of InGaAs/GaAs V-shaped quantum wires. J Opt Soc Am B, 1996, 13:1031 doi: 10.1364/JOSAB.13.001031
[22]	Lee J, Spector H N. Stark effect in the optical absorption in quantum wires. J Appl Phys, 2005, 97:043511 doi: 10.1063/1.1849430
[23]	Xia C X, Wei S Y, Spector H N. Franz-Keldysh effect in ZnO quantum wire. Physica E, 2010, 42:2065 doi: 10.1016/j.physe.2010.03.027
[24]	Yi G Y, Wei G Z, Wu H N. Transverse Stark effect in a rectangular semiconducting quantum wire. Phys Status Solidi B, 2007, 224:4651 doi: 10.1134/S1063739710060065
[25]	Zhang H, Wang X, Zhao J F, et al. The binding energy of a hydrogenic impurity in self-assembled double quantum dots. Chin Phys B, 2011, 20:127301 doi: 10.1088/1674-1056/20/12/127301
[26]	Riva C, Escorcia R A, Govorov A O, et al. Charged donors in quantum dots:finite difference and fractional dimensions results. Phys Rev B, 2004, 69:245306 doi: 10.1103/PhysRevB.69.245306

Transverse Stark effect in the optical absorption in a square semiconducting quantum wire

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