Báo cáo hóa học: " Electronic structures of GaAs/AlxGa1-xAs quantum double rings" ppt

Abstract In the framework of effective mass envelope function theory, the electronic structures of GaAs/Al x- Ga 1-x As quantum double rings (QDRs) are studied. Our model can be used to calculate the electronic structures of quantum wells, wires, dots, and the single ring. In calculations, the effects due to the different effective masses of electrons and holes in GaAs and Al x Ga 1-x As and the valence band mixing are considered. The energy levels of electrons and holes are calculated for different shapes of QDRs. The calculated results are useful in designing and fabricating the interrelated photoelectric devices. The single electron states presented here are useful for the study of the electron correlations and the effects of magnetic fields in QDRs. Keywords Electronic structures Æ GaAs Æ Quantum double rings Æ Nanostructures Æ Effective-mass theory Æ Band mixing PACS: 78.20.Bh Æ 78.66.Fd Introduction Growth of semiconductor nanostructures has attracted much attention due to their unique electronic and optical properties as well as potential applications in making electronic and optoelectronic devices. Recently, T. Mano et al. fabricated the self-assem- bled formation of concentric quantum double rings (QDRs) with high uniformity and excellent rotational symmetry using the dropletepitaxy technique [1]. They calculated the electronic energy levels using the effective mass approximation. For computational pur- poses, they assumed that the quantum rings have a rotational symmetry relative to the growth axis. Aside from this assumption, no adjustable parameters were used in the model. However, the valence band mixing was not considered in their calculations. We have studied the electronic states and valence band structures of the InAs/GaAs quantum single ring [2]. In this letter, using the effective-mass envelope- function theory, we will study the electron and hole states of QDRs. In our calculations, the effects due to the different effective masses of electrons and holes in GaAs and Al x Ga 1-x As and the valence band mixing are included. Our model can be used to calculate the electronic structures of quantum wells, wires, dots and the single ring. The single electron states are useful for the study of the electron correlations and the effects of magnetic fields on QDRs. Theoretical model Figure 1 shows the schematic plot of the GaAs/Al x Ga 1-x As QDRs. In the following, we choose z-direction of our coordinate system to be perpendicular to the plane of quantum rings. The QDRs are concentric. We suppose the inner radius and outer radius are R 1 , R 2 for the small ring and R 3 , R 4 for the large ring, respectively. The height of QDRs is l.IfR 1 = R 2 ,orR 3 = R 4 , the QDRs become quantum single ring. S S. Li (&) Æ J B. Xia State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, 912, Beijing 100083, People’s Republic of China e-mail: sslee@red.semi.ac.cn Nanoscale Res Lett (2006) 1:167–171 DOI 10.1007/s11671-006-9010-z 123 NANO EXPRESS Electronic structures of GaAs/Al x Ga 1-x As quantum double rings Shu-Shen Li Æ Jian-Bai Xia Published online: 11 August 2006 Ó to the authors 2006 According to Burt and Foreman’s effective-mass the- ory and taking into account the difference of the effec- tive-masses between GaAs and Al x Ga 1-x As [3, 4], the electron Hamiltonian can be written as (neglecting the second- and higher-order terms in the approximation) H e ¼ P 1 2m à e ðx; y; zÞ P þV e ðx; y; zÞ: ð1Þ In the above equation, m à e ðx;y;zÞ¼ m à 1 R 2 1 q 2 R 2 2 orR 2 3 q 2 R 2 4 ; andjzj l; m à 2 others; & ð2Þ V e ðx;y;zÞ¼ 0 R 2 1 q 2 R 2 2 orR 2 3 q 2 R 2 4 ; andjzj l; E c others; & ð3Þ where q 2 ¼ x 2 þ y 2 , and m 1 * and m 2 * are the effective electron masses in GaAs and Al x Ga 1-x As, respectively. E c is the conduction band offset between GaAs and Al x Ga 1-x As. The electron Schro ¨ dinger equation is H e W e ðr e Þ¼E e W e ðr e Þ: ð4Þ Using the periodic boundary condition, we assume that the electron wave functions have the following forms W e ðrÞ¼ 1 L 3=2 X n x n y n z a n x n y n z e iðk nx xþk ny yþk nz zÞ ; ð5Þ with k ni ¼ k i þ n i K, n i ¼ 0; Æ1; Æ2; , and i = x,y,z; K =2p/L, r =(x, y, z). L denotes the periods of the large units. The matrix elements of Hamiltonian (1) for Eq. 5 can be written as "h 2 2m à 2 d þ "h 2 2m à 12 S i S j ! k nx k 0 nx þ k ny k 0 ny þ k nz k 0 nz  þ d À S i S j ÀÁ E c ; ð6Þ where k ni ¢ = k i + n i ¢ K, and d ¼ 1 for n x ¼ n 0 x ; n y ¼ n 0 y ; and n z ¼ n 0 z ; 0 otherwise; & ð7Þ "h 2 2m à 12 ¼ "h 2 2m à 1 À "h 2 2m à 2 ; ð8Þ S i ¼ l=L; n z ¼ n à z ; sin½pðn z Àn 0 z Þl=L pðn z Àn 0 z Þ ; n z 6¼ n 0 z ; ( ð9Þ S j ¼ p½ðR 2 2 ÀR 2 1 ÞþðR 2 4 ÀR 2 3 Þ=L 2 ; n x ¼n 0 x and n y ¼n 0 y ; ðF 2 ÀF1 þF 4 ÀF 3 Þ=ðkLÞ; n x 6¼n 0 x or n y 6¼n 0 y : & ð10Þ In the above equation, F i ¼ R i J 1 ðkKR i Þwith i = 1, 2,3,4; J 1 is the first-order Bessel function J 1 ðxÞ¼ x 2p R p 0 cos xcoshðÞsin 2 h dh; and k ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n x À n 0 x ÀÁ 2 þ n y À n 0 y  2 r . Therefore, we can calculate the electronic states from Eq. 6. Fig. 1 Schematic plot of the GaAs/Al x Ga 1-x As quantum double rings 168 Nanoscale Res Lett (2006) 1:167–171 123 For the hole states, the hole effective mass Hamil- tonian can be written as [4] H h ¼ 1 2m 0 P þ R ÀQ À 0 R y P À C y ÀQ y þ ÀQ y À CP À ÀR 0 ÀQ þ ÀR y P þ 2 6 6 4 3 7 7 5 þ V h ; ð11Þ where V h ðrÞ¼ 0 R 2 1 q 2 R 2 2 or R 2 3 q 2 R 2 4 ; and jzj l; V h0 otherwise; & ð12Þ P Æ ¼ p x ðc 1 Æc 2 Þp x þp y ðc 1 Æc 2 Þp y þp z ðc 1 Ç2c 2 Þp z ; Q Æ ¼ 2 ffiffiffi 3 p ðp x Æip y Þrp z þp z pðp x Æip y Þ Âà ; R ¼ ffiffiffi 3 p ðp x þip y Þlðp x þip y ÞÀðp x Àip y Þcðp x Àip y Þ Âà ; C ¼ 2p z ðr ÀpÞðp x Àip y ÞÀ2ðp x Àip y Þðr ÀpÞp z ; ð13Þ and r ¼ðÀ1 Àc 1 þ 2c 2 þ 6c 3 Þ=6; p ¼ð1 þ c 1 À 2c 2 Þ=6; c ¼ðc 2 þ c 3 Þ=2; l ¼Àðc 2 À c 3 Þ=2: ð14Þ Here c 1 , c 2 ,andc 3 are functions of x, y, and z, c 1 ; c 2 ; c 3 ¼ c 11 ; c 12 ; c 13 for R 2 1 q 2 R 2 2 or R 2 3 q 2 R 2 4 ; and jzj l; c 21 ; c 22 ; c 23 otherwise: 8 < : ð15Þ The notations c 11 , c 12 , c 13 and c 21 , c 22 , c 23 are the Luttinger effective mass parameters of GaAs, Al x Ga 1- x As materials, respectively; and m 0 is the free electron mass. The hole envelope function equation is H h W h ¼ E h W h : ð16Þ Using the normalized plane-wave expansion method [5], we assume that the hole-wave functions have the following form: W h ðr h Þ¼ 1 L 3=2 X n x n y n z a n x n y n z b n x n y n z c n x n y n z d n x n y n z 2 6 6 4 3 7 7 5 e ik nx xþk ny yþk nz z ðÞ : ð17Þ The matrix elements of Hamiltonian (11) for Eq. 17 can be written as ðP Æ Þ n x n y n z ;n 0 x n 0 y n 0 z ¼ðc 1 Æ d þc 2 Æ S i S j Þðk nx k 0 nx þk ny k 0 ny Þ þðc 3 Æ d þc 4 Æ S i S j Þðk nz k 0 nz Þ; ðQ Æ Þ n x n y n z ;n 0 x n 0 y n 0 z ¼2 ffiffiffi 3 p ðr 2 Àd 2 Þd½ f Àðr 1 Àd 1 Àr 2 þd 2 ÞS i S j ÃÉ ðk 0 nx Æik 0 ny Þk 0 nz :þ p 2 d Àðp 1 Àp 2 ÞS i S j ÂÃÉ ðk nx Æik ny Þk 0 nz g; R n x n y n z ;n 0 x n 0 y n 0 z ¼ ffiffiffi 3 p l 2 d Àðl 1 Àl 2 ÞS i S j Âà ðk nx þik ny Þ È ðk 0 nx þik 0 ny ÞÀ c 2 d Àðc 1 Àc 2 ÞS i S j Âà ðk nx Àik ny Þðk 0 nx Àik 0 ny Þg; C n x n y n z ;n 0 x n 0 y n 0 z ¼2ðr 1 Àd 1 Àp 1 Àr 2 þd 2 þp 2 ÞS i S j ðk nx Àik ny Þk 0 nz Àðk 0 nx Àik 0 ny Þk nz hi ; ðV h Þ n x n y n z ;n 0 x n 0 y n 0 z ¼ðd ÀS i S j ÞV h0 ; with c 1 Æ ¼ c 21 Æ c 22 , c 2 Æ ¼ðc 11 Æ c 12 ÞÀc 1 Æ , c 3 Æ ¼ c 21 Ç 2c 22 , c 4 Æ ¼ðc 11 Ç 2c 12 ÞÀc 3 Æ , r i À d i ¼ðÀ1 Àc i1 þ 2c i2 þ6c i3 Þ=6, p i =(1+c i1 –2 c i2 )/6, c i =(c i2 + c i3 )/ 2, l i =–(c i2 – c i3 )/2, r i À d i À p i ¼ðÀ1 Àc i1 þ 2c i2 þ3c i3 Þ=3, and i = 1 or 2. Thus, the hole energy levels can be worked out from Eq. 18. Results and discussion We take the material parameters from Ref. 6. The aluminum proportion in Al x Ga 1-x As is taken to be x = 0.3, which equals the value for the experimental samples in Ref. 1. The effective masses and band gaps E g G (eV) are listed in Table 1. The conduction-band offset is assumed to be 65% of the band gap difference. We have calculated the electron and hole energy levels as functions of the radius of QDRs. In calcula- tions, we assume the height of QDRs to be l=3 nm. Figure 2a, b shows the electron and hole energy levels as a function of R 1 , respectively, for fixed Table 1 The effective masses and band gaps E g G (eV) of bulk GaAs and Al 0.3 Ga 0.7 As Material m à e ðm 0 Þ c 1 c 2 c 3 E g G (eV) GaAs 0.067 6.98 2.06 2.93 1.519 AlAs 0.15 3.76 0.82 1.42 3.099 Al 0.3 Ga 0.7 As 0.0919 6.014 1.688 2.477 1.993 Nanoscale Res Lett (2006) 1:167–171 169 123 R 2 = 6 nm, R 3 =8 nm, and R 4 =10 nm. From Fig. 2a, one may find that there is only one deep confined electronic energy level for the above structure param- eters. The anti-crossing is found near R 1 =4.5 nm for the second and the third electron energy levels. Figure 2b shows there are two confined hole energy levels for the above structure parameters. Figure 3a, b shows the electron and hole energy levels as a function of R 2 , respectively, for R 1 =4 nm, R 3 =8 nm, and R 4 =10 nm. The one and two confined electron energy levels is found for R 2 < and > 5.9 nm, respectively. The anti-crossing is found near the same R 2 =5.9 nm for the second and the third electron energy levels. The hole confined energy levels decrease monotonical as R 2 increases. Figure 4a, b shows the electron and hole energy levels as a function of R 3 , respectively, for R 1 =4 nm, R 2 =6 nm, and R 4 =10 nm. The one and two confined electron energy levels is found for R 3 < and > 8.1 nm, respectively. The anti-crossing is found near the same R 2 =8.1 nm for the second and the third electron energy levels. The only two confined hole energy levels is found for R 3 > 9.5 nm. Figure 5a, b shows the electron and hole energy levels as a function of R 4 , respectively, for R 1 =4 nm, R 2 =6 nm, and R 3 =8 nm. The one and two confined electron energy levels is found for R 4 < and > 10 nm, respectively. The anti-crossing is found near the same R 4 =10 nm for the second and the third electron energy levels. The only two confined hole energy levels is found for R 4 < 8.8 nm, Taking the structure parameters of the QDRs to be l = 3.5 nm, R 1 =10 nm, R 2 =35 nm, R 3 =40 nm, and R 4 =60 nm, the transition energies for the ground electron energy level transiting to the ground heavy- and light-hole energy levels were calculated to be 1.694, and 1.696 eV, respectively. These calculated 04 270 280 290 300 310 (a) 65321 04 140 150 160 (b) R 2 = 6 nm R 2 = 6 nm R 3 = 8 nm R 4 = 10 nm R 3 = 8 nm R 4 = 10 nm E h (meV) E e (meV) R 1 (nm) R 1 (nm) 65321 Fig. 2 The electron (a) and hole (b) energy levels as a function of R 1 for R 2 =6 nm, R 3 =8 nm, and R 4 =10 nm 47 280 290 300 310 (a) R 1 = 4 nm R 3 = 8 nm R 4 = 10 nm 865 47 135 140 145 150 155 160 165 E h (meV) E e (meV) R 2 (nm) R 2 (nm) (b) R 1 = 4 nm R 3 = 8 nm R 4 = 10 nm 865 Fig. 3 The electron (a) and hole (b) energy levels as a function of R 2 for R 1 =4 nm, R 3 =8 nm, and R 4 =10 nm 170 Nanoscale Res Lett (2006) 1:167–171 123 results are somewhat higher than the available exper- imental data in Ref. 1 for we have not included the binding energy of exciton. The exciton binding energy is estimated to be 15 meV from the difference between the theoretical values and experimental data. Summary In this paper, we have calculated the electronic states of GaAs/Al x Ga 1-x As QDRs. The model we proposed can be used to calculate the electronic states of quantum wells, wires, dots, and the single ring. The single electron states are useful for the study of the electron correlations and the effects of magnetic fields on QDRs. Our calculated results are useful in designing and fabricating the interrelated photoelec- tric devices. Acknowledgments This work was supported by the National Natural Science Foundation of China and the Special Founda- tions for State Major Basic Research Program of China (Grant No. G2001CB309500). References 1. T. Mano, T. Kuroda, S. Sanguinetti, T. Ochiai, T. Tateno, J. Kim, T. Noda, M. Kawabe, K. Sakoda, G. Kido, N. Kogu- chi, Nano Lett. 5, 425 (2005) 2. S S. Li, J B. Xia, J. Appl. Phys. 89, 3434 (2001) and J. Appl. Phys. 91, 3227 (2002) 3. M.G. Burt, J. Phys. Condens. Matter 4, 6651(1992) 4. B.A. Foreman, Phys. Rev. B 52, 12241(1995) 5. M.A. Cusack, P.R. Briddon, M. Jaros, Phys. Rev. B 54, 2300 (1996) 6. I. Vurgaftmana, J.R. Meyer, J. Appl. Phys. 89, 5815 (2001) 6910 280 290 300 310 E e (meV) R 3 (nm) (a) R 1 = 4 nm R 2 = 6 nm R 4 = 10 nm 87 6910 140 150 160 E h (meV) R 3 (nm) (b) R 1 = 4 nm R 2 = 6 nm R 4 = 10 nm 87 Fig. 4 The electron (a) and hole (b) energy levels as a function of R 3 for R 1 =4 nm, R 2 =6 nm, and R 4 =10 nm 8101112 280 290 300 310 E e (meV)E h (meV) R 4 (nm) R 4 (nm) (a) R 1 = 4 nm R 2 = 6 nm R 3 = 8 nm 9 8101112 140 145 150 155 160 165 (b) R 1 = 4 nm R 2 = 6 nm R 3 = 8 nm 9 Fig. 5 The electron (a) and hole (b) energy levels as a function of R 4 for R 1 =4 nm, R 2 =6 nm, and R 3 =8 nm Nanoscale Res Lett (2006) 1:167–171 171 123 . framework of effective mass envelope function theory, the electronic structures of GaAs/Al x- Ga 1-x As quantum double rings (QDRs) are studied. Our model can be used to calculate the electronic structures of. here are useful for the study of the electron correlations and the effects of magnetic fields in QDRs. Keywords Electronic structures Æ GaAs Æ Quantum double rings Æ Nanostructures Æ Effective-mass. calculate the electronic structures of quantum wells, wires, dots and the single ring. The single electron states are useful for the study of the electron correlations and the effects of magnetic