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Electric field and shape effect on the linear and nonlinear optical properties of multi-shell ellipsoidal quantum dots
Superlattices and Microstructures 94 (2016) 204e214
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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Electric field and shape effect on the linear and nonlinear optical properties of multi-shell ellipsoidal quantum dots L. Shi, Z.W. Yan* College of Science, Inner Mongolia Agricultural University, Hohhot, 010018, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 March 2016 Received in revised form 13 April 2016 Accepted 16 April 2016 Available online 19 April 2016
In the present work, the optical properties of GaAs/AlxGa1xAs/GaAs multi-shell ellipsoidal quantum dot heterostructures with a shallow hydrogenic impurity in the presence of an external electric field have been studied. The results show how the linear and nonlinear optical absorption coefficients and refraction index changes are changed by the variations of the size and shape of the multi-shell structure. Moreover, how the optical properties of this structure are affected by the electric field has also been shown. The physical reasons for the results have been discussed in detail. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Multi-shell ellipsoidal quantum dot Electric field Optical property
1. Introduction Linear and nonlinear optical absorption coefficients (ACs) and refractive index changes (RICs) are an important source of information about the optical properties of luminescent device, and therefore it has the potential for device research [1e3]. Furthermore, there is no doubt that the zero-dimensional quantum dots (QDs) with well-controlled shape and size have became one of the hottest topics in the area of the condensed matter and materials physics due to their distinctive electronic and optical properties, which shows a potential application in the electronic and optoelectronic devices. Therefore, the optical properties of QDs have attracted the considerable attention in the experimental and theoretical studies in recent years [4e7]. Development in growth technology has resulted in manufacturing of multi-shell quantum dot (MSQDs) structures, and it has attracted more attention than single-layered QDs because of some advantageous properties, such as lower Auger recombination coefficients, higher absorption cross-sections, wider range of absorption and emission spectra, and better photoluminescence properties [8,9], therefore MSODs show better optical performance. In particular, it has a considerable potential application in QD solar cell applications due to its advantageous properties mentioned above [10e12]. Because the shape of QDs lead to the formation of discrete energy levels, the optical properties of semiconductor QDs depend on the shape of QDs. In the theoretical works, it is customary to assume a spherical shape for the MSQDs, and many studies were focused on the optical properties of such structure [4,5]. But from physical point of view the consideration of ellipsoidal shape is actual due to unavoidable small deviations from spherical shape because of deformations during QD growth. Generally speaking, if the MSQD is weakly prolate (oblate), the problem can be solved within the frame work of perturbation theory. In order to modulate the properties of devices, the effect of electric field on the electronic states in QDs has been studied extensively by many authors in the past few years [13e15]. As we known, the electric field induces both a polarization of the carrier distribution and an energy shift of the quantum state to introduce a considerable change in the energy spectra of
* Corresponding author. E-mail address: [email protected] (Z.W. Yan). http://dx.doi.org/10.1016/j.spmi.2016.04.022 0749-6036/© 2016 Elsevier Ltd. All rights reserved.
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carriers, and this property is very useful for optoelectronic devices. Because applied electric field has an important influence on the carrier energy and carrier distribution, the optical properties of QDs also depend on the electric field. There are many works about the optical properties of QDs under the influence of electric field [16e18], but for multi-shell ellipsoidal quantum dot (MSEQD) structures it appears insufficient. Actually, the MSEQD is more sensitive to the electric field due to their multishell structures, and it is expected that the effect of electric field on the optical properties of MSEQDs will be more pronounced. In the present paper, the optical properties of GaAs/AlxGa1xAs/GaAs MSEQD heterostructures in the presence of an external electric field have been studied by using a variational approach. As is well known, the existence of impurity which is more close to the actual situation has a strong influence on the electric and optical properties of QDs via changing the quantized energy spectrum and probability density of the charge carriers. Therefore we also consider the effect of an oncenter shallow hydrogenic impurity on the optical properties of MSEQDs. In this study, the linear, third-order nonlinear, and total ACs and RICs under the influence of electric field are investigated and discussed. This paper is organized as follows. In Section 2, we present our model and explain the general theory. In Section 3, the numerical results and detailed discussions are given. Finally, the conclusions are presented in Section 4. 2. Theory 2.1. Hamiltonian and wave function In this study, a GaAs/AlxGa1xAs/GaAs MSEQD structure as seen in Fig. 1 is considered. The core region, barrier region and well region denoted by U1, U2 and U3, respectively. Let us indicate with a and c its semi-axis in the x-y plane and along the z axis, respectively. The equation of the QD surface is
X2 þ Y2 Z2 þ 2 ¼ 1: a2 c
(1)
In the effective mass approximation, the Hamiltonian of an electron bound to an on-center shallow hydrogenic impurity in the presence of an external electric field along the z-direction is
H0 ¼
Z2 2 V þ UðrÞ þ VðrÞ þ jejFr cos q; 2m
with
Fig. 1. Schematic diagrams of the MSEQD heterostructure and corresponding confinement potentials.
(2)
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8 < 0; r U1 and U2 < r U3 m1 ; r U1 and U2 < r U3 VðrÞ ¼ Vc ; U1 < r U2 ; m¼ ; m2 ; U1 < r U2 : ∞; r > U3
(3)
where m is the effective mass of electron in different regions; U(r) is the Coulomb potential between the electron and impurity; V(r) is the step-like confinement potential; F is the strength of the electric field along the z-direction; q is the angle . between the electronic position vector r and the electric field direction. For the weakly prolate or oblate ellipsoidal QD (ellipsoid is only slightly different from sphere) the problem may be solved within the framework of perturbation theory. It is useful to change the variables as follows [19e21]:
X¼
1=3 a a c x; Y ¼ y; Z ¼ z; R ¼ a2 c : R R R
(4)
This change transforms the MSEQD into a sphere with the same volume. In the transformed structure, the core radius is R1, the barrier shell width is Bw ¼ R2 R1, and the well width is Ww ¼ R3 R2. After the transformations, the Hamiltonian of the system can be transformed to the form:
H ¼ H0 þ H 0 þ V 0 ðrÞ;
(5)
Z2 2 V þ U0 ðrÞ þ V0 ðrÞ þ jejFr cos q; 2m 1 c a _2 c2 þ a2 þ ac_2 e2 c a 1 c2 þ a2 þ ac 2 2ðc aÞ p jejFr cos q: H0 ¼ þ p cos q z 3m a 3 3a 4pε0 εr c c2 3a2 H0 ¼
(6) (7)
The confinement potential V0(r) is changed to
8 < 0; V0 ðrÞ ¼ Vc ; : ∞;
r R1 and R2 < r R3 : R1 < r R2 r > R3
(8)
The last term on the right-hand-side of Eq. (5) is the non-spherical part of the confinement potential, which can be neglected [19]. Considering the dielectric mismatch between the different regions, the Coulomb potential U0(r) can be written as [22,23].
8 e2 e2 ε1 e2 ε2 e2 ε3 > > > þ 1 þ 1 þ 1 ; > > 4pε0 ε1 r 4pε0 ε1 R1 ε2 4pε0 ε2 R2 ε3 4pε0 ε3 R3 ε4 > > > > < e2 e2 ε e2 ε U0 ðrÞ ¼ þ 1 2 þ 1 3 ; > 4pε0 ε2 r 4pε0 ε2 R2 ε3 4pε0 ε3 R3 ε4 > > > > > 2 2 > > e e ε > : þ 1 3 ; 4pε0 ε3 r 4pε0 ε3 R3 ε4
r R1 R1 < r R2 :
(9)
R2 < r R3
Eq. (9) includes the direct electron-ion interaction in different regions and the existence of the polarized surface charges at different region boundaries. ε1, ε2 and ε3 are the static dielectric constants of the core, barrier and well materials respectively. It is necessary to note that the equation undergoes the greatest change at ε4 ¼ 1, i.e., when the MSEQD structure is in vacuum. We introduce now an ellipticity constant b ¼ 1 a/c z c/a 1 to express the degree of ellipticity, where b < 0 means the oblate shape, b > 0 means the prolate shape, and b ¼ 0 means the spherical shape. For the weakly prolate or oblate MSEQDs, we have jbj < < 1 and thus H 0 may be considered as a perturbation. Therefore, the spherical part plays the main role and we can solve the problem conveniently by using the spherical coordinates. €dinger equation, two cases need to be distinguished. In the region where the electronic ground state To solve the Schro energy E0 > Vc, we choose the following trial function [24,25].
8 A1 j k r þ B1 nl knl;1 r f ðrÞ; > > < l nl;1 A j k r þ B2 nl knl;2 r f ðrÞ; fðrÞ ¼ Rnl ðrÞf ðrÞ ¼ 2 l nl;2 > > : A3 jl knl;3 r þ B3 nl knl;3 r f ðrÞ; 0
r R1 R1 < r R2 ; R2 < r R3 r > R3
(10)
L. Shi, Z.W. Yan / Superlattices and Microstructures 94 (2016) 204e214
f ðrÞ ¼
el1 r eh1 r cos q r cos qel2 r eh2 r cos q
for the ground state : for the first excited state
207
(11)
In the region where the electronic ground state energy E0 < Vc, we choose the following trial function
8 A1 j k r þ B1 nl knl;1 r f ðrÞ; > > > l ðþÞnl;1 < ðÞ A2 hl iknl;2 r þ B2 hl iknl;2 r f ðrÞ; fðrÞ ¼ Rnl ðrÞf ðrÞ > > A3 jl knl;3 r þ B3 nl knl;3 r f ðrÞ; > : 0 l r h r cos q for the ground state e 1 e 1 : f ðrÞ ¼ r cos qel2 r eh2 r cos q for the first excited state
r R1 R1 < r R2 R2 < r R3 r > R3
;
(12)
(13)
The coefficients in Eqs. (10) and (12) are as follows:
knl;1 ¼
Z2
knl;2 ¼
knl;3 ¼
2m1 E0
1=2 ;
2m2 jV0 E0 j Z2
p R3
(14) 1=2 ;
(15)
;
(16) ðþÞ
ðÞ
where jl(x), nl(x), hl ðxÞ and hl ðxÞ are the spherical Bessel function, Neumann function, and Hankel functions of the first and the second kinds. A1, A2, A3, B1, B2 and B3 are normalized constants. The boundary conditions for these functions are [26].
Rnl;1 ðR1 Þ ¼ Rnl;2 ðR1 Þ; Rnl;2 ðR2 Þ ¼ Rnl;3 ðR2 Þ; Rnl;3 ðR3 Þ ¼ 0;
(17)
1 dRnl;1 1 dRnl;2 ¼ ; m1 dr r¼R1 m2 dr r¼R1
(18)
ZR1
r 2 R*nl;1 ðrÞRnl;1 ðrÞdr
ZR2 þ
r2 R*nl;2 ðrÞRnl;2 ðrÞdr
R1
0
1 dRnl;2 1 dRnl;3 ¼ ; m2 dr r¼R2 m3 dr r¼R2 ZR3 þ
r 2 R*nl;3 ðrÞRnl;3 ðrÞdr ¼ 1:
(19)
R2
Because the wave function is limited when r/0, B1 ¼ 0. It is obvious that A1, A2, A3, B2, B3 and E0 can be obtained by solving the Eqs. (17)e(19). It is necessary to explain that l1(l2) and h1(h2) are variational parameters for the ground (excited) state to indicate the degree of spatial correlation between electron and impurity and the influence of the electric field, respectively. Then, the ground and excited state energies of the impurity state can be obtained by minimizing the expectation energies with respect to l1, h1 and l2, h2
〈4ðrÞjHj4ðrÞ〉 ; l1 ;h1 〈4ðrÞj4ðrÞ〉
(20)
〈4ðrÞjHj4ðrÞ〉 : 〈4ðrÞj4ðrÞ〉
(21)
E1 ¼ min E2 ¼ min l2 ;h2
2.2. Optical properties of the system In this work, an optical transition between the ground state and first excited state has been considered. The linear and third-order nonlinear ACs and RICs of the impurity state in a MSEQD can be obtained by a density matrix approach and a perturbation expansion method. The linear AC að1Þ ðuÞ and the third-order nonlinear AC að3Þ ðu; IÞ are calculated by Refs. [7,16,17,27].
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Fig. 2. Probability of finding the electron in ground and first excited states without electric field in various regions as a function of the core radius R1 for b ¼ 0 and Ww ¼ Bw ¼ 2 nm (a), or of the barrier shell width Bw for b ¼ 0, Ww ¼ 2 nm, and R1 ¼ 3 nm (b), or of the well width Ww for b ¼ 0, Bw ¼ 2 nm, and R1 ¼ 3 nm (c), or of the ellipticity constant b for R1 ¼ 3 nm and Ww ¼ Bw ¼ 2 nm (d). The Al content is chosen as x ¼ 0.3.
að1Þ ðuÞ ¼ u
ð3Þ
rffiffiffiffiffi
m
jM21 j2 sV ZG12
εR ðE21 ZuÞ2 þ ðZG12 Þ2
a ðu; IÞ ¼ u
rffiffiffiffiffi
m
εR
I 2ε0 nr c
h
;
(22) 2
jM21 j2 sV ZG12
i2 44jM21 j
2
ðE21 ZuÞ þ ðZG12 Þ i3 2 4E Zu þ Z2 u2 G2 jM22 M11 j2 3E21 21 12 5; 2 þ ðZG Þ2 E21 12 2
2
h
(23)
the total AC is defined as
aðuÞ ¼ að1Þ ðuÞ þ að3Þ ðu; IÞ:
(24)
The linear and the third-order nonlinear RICs are given by Refs. [7,17].
Dnð1Þ ðuÞ nr
¼
" # 1 E21 Zu 2 s ; jM21 j V 2n2r ε0 ðE21 ZuÞ2 þ ðZG12 Þ2
(25)
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Dnð3Þ ðu; IÞ nr
209
h h ðM22 M11 Þ2 n 2 ðE21 ZuÞ E21 ðE21 ZuÞ i2 4ðE21 ZuÞjM21 j 2 2 E21 þ ðZG12 Þ ðE21 ZuÞ2 þ ðZG12 Þ2 i oi ðZG12 Þ2 ðZG12 Þ2 ð2E21 ZuÞ
mc ¼ 3 jM21 j2 h 4nr ε0
sV I
(26) the total RIC is defined as
Dnðu; IÞ nr
¼
Dnð1Þ ðu; IÞ nr
þ
Dnð3Þ ðu; IÞ nr
;
(27)
where m is the magnetic susceptibility, sV is the electron density, nr is the refractive index of dot material, ħu is the incident photon energy, εR ¼ n2r ε0 is the real part of the permittivity, I is the incident light intensity, G12 is the relaxation rate for the ground and excited states, E21 ¼ E2E1 is the energy difference between two different electronic states, and M21 is the transition matrix element between the initial and final states. Here, we have chosen a polarized electromagnetic radiation in the z-direction, and hence the transition matrix element is defined as
M21 ¼ j〈42 jezj41 〉j:
(28)
3. Results and discussion In this study, the numerical calculations have been carried out for a GaAs/AlxGa1xAs/GaAs MSEQD system by using the following material parameters: sV ¼ 3.0 1016 cm3, nr ¼ 3.2, T12 ¼ 0.2 ps, G12 ¼ 1/T12 [27]. The effective mass and static dielectric are: m1 ¼ 0.067m0, m2 ¼ (0.067 þ 0.083)m0 (m0 is the mass of a free electron), and ε1 ¼ ε3 ¼ 13.18, ε2 ¼ 13.18 e 3.12x [28]. The band offset for the GaAs/AlxGa1xAs is given as V0 ¼ 0.7(1247x) meV [28]. In order to understand the problem more clearly in the next discussion, firstly the probability of finding the electron in the ground and first excited states in different regions are shown in Fig. 2. The barrier shell, well width and core radius for each figure are chosen to allow the electron tunneling between each region. Fig. 2(a) shows that for the very small core radius, the electron in the ground state is mainly confined in the core region, but the electron in the excited state is mainly confined in the
Fig. 3. Transition energy as a function of electric field for various R1 values with b ¼ 0 and Ww ¼ Bw ¼ 2 nm, or for various Bw values with b ¼ 0, Ww ¼ 2 nm, and R1 ¼ 3 nm, or for various Ww values with b ¼ 0, Bw ¼ 2 nm, and R1 ¼ 3 nm, or for various ellipticity constant b with R1 ¼ 3 nm, Ww ¼ Bw ¼ 2 nm. The Al content is chosen as x ¼ 0.3.
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Fig. 4. The linear (dotted lines), third-order nonlinear (dashed lines) and total (solid lines) ACs (a) and RICs (b) with and without electric field as a function of photon energy ħu for various R1 and b values with Ww ¼ Bw ¼ 2 nm, x ¼ 0.3 and I ¼ 0.3 MW/cm2. Red, dark and blue lines correspond to b ¼ 0.2, 0, 0.2, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
barrier shell region. As expected, as the core radius increases the penetration of the electron wave function into shell and well regions is weakened due to the impurity, therefore the electron in the excited state is also mainly confined in the core region. Fig. 2(b) shows that the electron in the ground state is mainly confined in the core region, and the variation of barrier shell thickness has little effect on the ground state. By contrast, the electron in the excited state is mainly confined in the shell region, and this confinement is obviously enhanced as the shell width increases. Fig. 2(c) shows that as the well width increases the penetration of the electron wave functions into well region is greatly increased but that into core and shell regions is decreased, and finally the electron in ground and excited states is mainly confined in the well region. Fig. 2(d) shows that the variation of the b value has little effect on the probability. Because the electric field and shape have an important influence on the transition energies, Fig. 3 shows the transition energies as a function of electric field under different conditions. It is found that the increasing electric field decreases the transition energies, and this behavior is more pronounced for the larger core radius or shell thickness or well width. In particular, we found that the influence of electric field is more pronounced for larger b values, which means that the prolate case is more sensitive to the electron field compared with the oblate case. This is because the prolate case expands in the zdirection size of the MSEQD, which is just the direction of the electric field, and therefore the influence of the electric field is enhanced. The ACs and RICs with and without the electric field as a function of the photon energy for different core radii and b values are plotted in Fig. 4. It is seen that the increasing core radius and/or electric field and/or b value red shifts the peak positions of linear, nonlinear and total ACs and RICs, and the red shift caused by the electric field is more pronounced for the larger core radius, this is because the increasing core radius and/or electric field and/or b value decreases the transition energy, and the
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211
Fig. 5. The linear (dotted lines), third-order nonlinear (dashed lines) and total (solid lines) ACs (a) and RICs (b) with and without electric field as a function of photon energy ħu for various Bw and b values with R1 ¼ 3 nm, Ww ¼ 2 nm, x ¼ 0.3 and I ¼ 0.3 MW/cm2. Red, dark and blue lines correspond to b ¼ 0.2, 0, 0.2, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
influence of electric field is more pronounced for the larger core radius as shown in Fig. 3. However, as expected, the red shift caused by b value is more pronounced for the smaller core radius, by reason of the effect of shape will become more and more weak as the size of MSEQD such as core radius, shell thickness and well width is increased. In addition, it is found that the peak intensities of nonlinear ACs, linear and nonlinear RICs increase with the increasing core radius, but this behavior is insensitive to the linear ACs. It can be explained that because the increasing core radius suppresses the tunneling effect and therefore the electron is mainly confined in core region as can be seen in Fig. 2(a), the overlapping of the wave functions of the initial and final states is increased, and the peak intensities of nonlinear ACs, linear and nonlinear RICs are increased. However, the peak intensity of linear ACs also depends on the transition energy E21 besides the overlapping of the wave functions. In contrast to the increasing overlapping of the wave functions, the transition energy experiences a decrease with increasing core radius as shown in Fig. 3. The competition effect between these two factors causes the linear ACs is insensitive to the variation of core radius. Fig. 5 shows the ACs and RICs with and without electric field as a function of the photon energy for different shell thickness. It shows that the increasing shell thickness and/or electric field and/or b value red shifts the peak positions of linear, nonlinear and total ACs and RICs, and the red shift caused by the electric field is more pronounced for the larger shell thickness. It can be also explained by Fig. 3 which shows that the increasing shell thickness and/or electric field and/or b value decreases the transition energy, and the influence of electric field on the transition energy is more pronounced for the larger shell thickness. Similar to Fig. 4, the red shift caused by b value is more pronounced for the smaller shell thickness. In contrast to Fig. 4, it is found that the increasing shell thickness decreases the peak intensities of linear and nonlinear ACs, but almost no effect on RICs. It can be explained that the electron in the ground state is mainly confined in the core region, but the increasing shell thickness obviously facilitates the electron in the excited state confined in the shell region as shown in Fig. 2(b).
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Fig. 6. The linear (dotted lines), third-order nonlinear (dashed lines) and total (solid lines) ACs (a) and RICs (b) with and without electric field as a function of photon energy ħu for various Ww and b values with R1 ¼ 3 nm, Bw ¼ 2 nm, x ¼ 0.3 and I ¼ 0.2 MW/cm2. Red, dark and blue lines correspond to b ¼ 0.2, 0, 0.2, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Therefore, the overlapping of the wave functions of the initial and final states is decreased, and their peak intensities are decreased. However, the variation of the overlapping of the wave functions is too small to obviously change the peak intensities of RICs, and therefore lead to the results as shown. The ACs and RICs with and without electric field as a function of the photon energy for different well widths are plotted in Fig. 6. It shows that the increasing well width and/or electric field and/or b value red shifts the peak positions of linear, nonlinear and total ACs and RICs, and the red shift caused by the electric field is more pronounced for the larger well width, but that caused by b value is more pronounced for the smaller well width. It can be also explained by Fig. 3 which shows that the increasing well width and/or electric field and/or b value decreases the transition energy, and the influence of electric field on the transition energy is more pronounced for the larger well width. In addition, it is found that the increasing well width obviously increases the peak intensities of linear and nonlinear RICs and nonlinear ACs, but it has little effect on the linear ACs. The reason can be found in Fig. 2(c), which shows that the electron in ground and excited states is more and more confined in the well region with the increasing well width, and therefore the overlapping of the wave functions of the initial and final states is increased, which opposites to the variation of transition energy. The competition effect between these two factors causes the linear ACs is insensitive to the variation of well width. In particular, it can be seen that the peak of total AC for Ww ¼ 6 nm under the electric field obviously splits into two peaks, and the split is enhanced with increasing b. Finally, the ACs and RICs with and without electric field as a function of the photon energy for different Al contents are presented in Fig. 7. It shows that the increasing Al content blue shifts the peak positions of linear, nonlinear and total ACs and RICs, but the increasing electric field and/or b value red shift them. This is because the transition energy is increased by the increasing Al content due to the quantum size effect, but reduced by the electric field. In addition, it is found that the peak intensities of linear ACs almost no change with the increasing Al content, but the linear and nonlinear RICs and nonlinear ACs
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213
Fig. 7. The linear (dotted lines), third-order nonlinear (dashed lines) and total (solid lines) ACs (a) and RICs (b) with and without electric field as a function of photon energy ħu for various x and b values with R1 ¼ 6 nm, Ww ¼ Bw ¼ 2 nm, and I ¼ 0.3 MW/cm2.
decrease monotonically with it. It is because the increasing Al content increases the energy level difference due to the quantum size effect, the overlapping of the wave functions of the initial and final states is decreased, which counteracts the increasing transition energy. 4. Conclusions In summary, the linear and nonlinear optical properties of a GaAs/AlxGa1xAs/GaAs MSEQD under the influence of an external field are studied. The results indicate that the electric field red shifts the peak positions of the ACs and RICs, and its
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influence is more pronounced for the large regions, i.e. core, shell and well ones. By contrast, the increasing Al content blue shifts the peak positions. In addition, we observe that the shape of MSEQD also has an important influence on the ACs and RICs. Compared with the spherical case (b ¼ 0), the prolate case (b > 0) red shifts the peak positions of the ACs and RICs, but the oblate case (b < 0) blue shifts them. This study provides an understanding of the optical properties of MSQD structures and how the optical properties are affected by the electric field. We hope that the present theory can stimulate further studies related to the optical properties of MSEQDs, and help experimenters use electric field to modulate the optical properties of MSEQDs. Acknowledgments This work was supported by the National Natural Science Foundation of PR China (Project No. 11364028), the Major Projects of Natural Science Foundation of Inner Mongolia (Project 2013ZD02), the Doctoral Staring up Foundation of Inner Mongolia Agricultural University (Project BJ2013B-2), the Grassland talent project. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Electric field and shape effect on the linear and nonlinear optical properties of multi-shell ellipsoidal quantum dots L. Shi, Z.W. Yan* College of Science, Inner Mongolia Agricultural University, Hohhot, 010018, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 March 2016 Received in revised form 13 April 2016 Accepted 16 April 2016 Available online 19 April 2016
In the present work, the optical properties of GaAs/AlxGa1xAs/GaAs multi-shell ellipsoidal quantum dot heterostructures with a shallow hydrogenic impurity in the presence of an external electric field have been studied. The results show how the linear and nonlinear optical absorption coefficients and refraction index changes are changed by the variations of the size and shape of the multi-shell structure. Moreover, how the optical properties of this structure are affected by the electric field has also been shown. The physical reasons for the results have been discussed in detail. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Multi-shell ellipsoidal quantum dot Electric field Optical property
1. Introduction Linear and nonlinear optical absorption coefficients (ACs) and refractive index changes (RICs) are an important source of information about the optical properties of luminescent device, and therefore it has the potential for device research [1e3]. Furthermore, there is no doubt that the zero-dimensional quantum dots (QDs) with well-controlled shape and size have became one of the hottest topics in the area of the condensed matter and materials physics due to their distinctive electronic and optical properties, which shows a potential application in the electronic and optoelectronic devices. Therefore, the optical properties of QDs have attracted the considerable attention in the experimental and theoretical studies in recent years [4e7]. Development in growth technology has resulted in manufacturing of multi-shell quantum dot (MSQDs) structures, and it has attracted more attention than single-layered QDs because of some advantageous properties, such as lower Auger recombination coefficients, higher absorption cross-sections, wider range of absorption and emission spectra, and better photoluminescence properties [8,9], therefore MSODs show better optical performance. In particular, it has a considerable potential application in QD solar cell applications due to its advantageous properties mentioned above [10e12]. Because the shape of QDs lead to the formation of discrete energy levels, the optical properties of semiconductor QDs depend on the shape of QDs. In the theoretical works, it is customary to assume a spherical shape for the MSQDs, and many studies were focused on the optical properties of such structure [4,5]. But from physical point of view the consideration of ellipsoidal shape is actual due to unavoidable small deviations from spherical shape because of deformations during QD growth. Generally speaking, if the MSQD is weakly prolate (oblate), the problem can be solved within the frame work of perturbation theory. In order to modulate the properties of devices, the effect of electric field on the electronic states in QDs has been studied extensively by many authors in the past few years [13e15]. As we known, the electric field induces both a polarization of the carrier distribution and an energy shift of the quantum state to introduce a considerable change in the energy spectra of
* Corresponding author. E-mail address: [email protected] (Z.W. Yan). http://dx.doi.org/10.1016/j.spmi.2016.04.022 0749-6036/© 2016 Elsevier Ltd. All rights reserved.
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205
carriers, and this property is very useful for optoelectronic devices. Because applied electric field has an important influence on the carrier energy and carrier distribution, the optical properties of QDs also depend on the electric field. There are many works about the optical properties of QDs under the influence of electric field [16e18], but for multi-shell ellipsoidal quantum dot (MSEQD) structures it appears insufficient. Actually, the MSEQD is more sensitive to the electric field due to their multishell structures, and it is expected that the effect of electric field on the optical properties of MSEQDs will be more pronounced. In the present paper, the optical properties of GaAs/AlxGa1xAs/GaAs MSEQD heterostructures in the presence of an external electric field have been studied by using a variational approach. As is well known, the existence of impurity which is more close to the actual situation has a strong influence on the electric and optical properties of QDs via changing the quantized energy spectrum and probability density of the charge carriers. Therefore we also consider the effect of an oncenter shallow hydrogenic impurity on the optical properties of MSEQDs. In this study, the linear, third-order nonlinear, and total ACs and RICs under the influence of electric field are investigated and discussed. This paper is organized as follows. In Section 2, we present our model and explain the general theory. In Section 3, the numerical results and detailed discussions are given. Finally, the conclusions are presented in Section 4. 2. Theory 2.1. Hamiltonian and wave function In this study, a GaAs/AlxGa1xAs/GaAs MSEQD structure as seen in Fig. 1 is considered. The core region, barrier region and well region denoted by U1, U2 and U3, respectively. Let us indicate with a and c its semi-axis in the x-y plane and along the z axis, respectively. The equation of the QD surface is
X2 þ Y2 Z2 þ 2 ¼ 1: a2 c
(1)
In the effective mass approximation, the Hamiltonian of an electron bound to an on-center shallow hydrogenic impurity in the presence of an external electric field along the z-direction is
H0 ¼
Z2 2 V þ UðrÞ þ VðrÞ þ jejFr cos q; 2m
with
Fig. 1. Schematic diagrams of the MSEQD heterostructure and corresponding confinement potentials.
(2)
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8 < 0; r U1 and U2 < r U3 m1 ; r U1 and U2 < r U3 VðrÞ ¼ Vc ; U1 < r U2 ; m¼ ; m2 ; U1 < r U2 : ∞; r > U3
(3)
where m is the effective mass of electron in different regions; U(r) is the Coulomb potential between the electron and impurity; V(r) is the step-like confinement potential; F is the strength of the electric field along the z-direction; q is the angle . between the electronic position vector r and the electric field direction. For the weakly prolate or oblate ellipsoidal QD (ellipsoid is only slightly different from sphere) the problem may be solved within the framework of perturbation theory. It is useful to change the variables as follows [19e21]:
X¼
1=3 a a c x; Y ¼ y; Z ¼ z; R ¼ a2 c : R R R
(4)
This change transforms the MSEQD into a sphere with the same volume. In the transformed structure, the core radius is R1, the barrier shell width is Bw ¼ R2 R1, and the well width is Ww ¼ R3 R2. After the transformations, the Hamiltonian of the system can be transformed to the form:
H ¼ H0 þ H 0 þ V 0 ðrÞ;
(5)
Z2 2 V þ U0 ðrÞ þ V0 ðrÞ þ jejFr cos q; 2m 1 c a _2 c2 þ a2 þ ac_2 e2 c a 1 c2 þ a2 þ ac 2 2ðc aÞ p jejFr cos q: H0 ¼ þ p cos q z 3m a 3 3a 4pε0 εr c c2 3a2 H0 ¼
(6) (7)
The confinement potential V0(r) is changed to
8 < 0; V0 ðrÞ ¼ Vc ; : ∞;
r R1 and R2 < r R3 : R1 < r R2 r > R3
(8)
The last term on the right-hand-side of Eq. (5) is the non-spherical part of the confinement potential, which can be neglected [19]. Considering the dielectric mismatch between the different regions, the Coulomb potential U0(r) can be written as [22,23].
8 e2 e2 ε1 e2 ε2 e2 ε3 > > > þ 1 þ 1 þ 1 ; > > 4pε0 ε1 r 4pε0 ε1 R1 ε2 4pε0 ε2 R2 ε3 4pε0 ε3 R3 ε4 > > > > < e2 e2 ε e2 ε U0 ðrÞ ¼ þ 1 2 þ 1 3 ; > 4pε0 ε2 r 4pε0 ε2 R2 ε3 4pε0 ε3 R3 ε4 > > > > > 2 2 > > e e ε > : þ 1 3 ; 4pε0 ε3 r 4pε0 ε3 R3 ε4
r R1 R1 < r R2 :
(9)
R2 < r R3
Eq. (9) includes the direct electron-ion interaction in different regions and the existence of the polarized surface charges at different region boundaries. ε1, ε2 and ε3 are the static dielectric constants of the core, barrier and well materials respectively. It is necessary to note that the equation undergoes the greatest change at ε4 ¼ 1, i.e., when the MSEQD structure is in vacuum. We introduce now an ellipticity constant b ¼ 1 a/c z c/a 1 to express the degree of ellipticity, where b < 0 means the oblate shape, b > 0 means the prolate shape, and b ¼ 0 means the spherical shape. For the weakly prolate or oblate MSEQDs, we have jbj < < 1 and thus H 0 may be considered as a perturbation. Therefore, the spherical part plays the main role and we can solve the problem conveniently by using the spherical coordinates. €dinger equation, two cases need to be distinguished. In the region where the electronic ground state To solve the Schro energy E0 > Vc, we choose the following trial function [24,25].
8 A1 j k r þ B1 nl knl;1 r f ðrÞ; > > < l nl;1 A j k r þ B2 nl knl;2 r f ðrÞ; fðrÞ ¼ Rnl ðrÞf ðrÞ ¼ 2 l nl;2 > > : A3 jl knl;3 r þ B3 nl knl;3 r f ðrÞ; 0
r R1 R1 < r R2 ; R2 < r R3 r > R3
(10)
L. Shi, Z.W. Yan / Superlattices and Microstructures 94 (2016) 204e214
f ðrÞ ¼
el1 r eh1 r cos q r cos qel2 r eh2 r cos q
for the ground state : for the first excited state
207
(11)
In the region where the electronic ground state energy E0 < Vc, we choose the following trial function
8 A1 j k r þ B1 nl knl;1 r f ðrÞ; > > > l ðþÞnl;1 < ðÞ A2 hl iknl;2 r þ B2 hl iknl;2 r f ðrÞ; fðrÞ ¼ Rnl ðrÞf ðrÞ > > A3 jl knl;3 r þ B3 nl knl;3 r f ðrÞ; > : 0 l r h r cos q for the ground state e 1 e 1 : f ðrÞ ¼ r cos qel2 r eh2 r cos q for the first excited state
r R1 R1 < r R2 R2 < r R3 r > R3
;
(12)
(13)
The coefficients in Eqs. (10) and (12) are as follows:
knl;1 ¼
Z2
knl;2 ¼
knl;3 ¼
2m1 E0
1=2 ;
2m2 jV0 E0 j Z2
p R3
(14) 1=2 ;
(15)
;
(16) ðþÞ
ðÞ
where jl(x), nl(x), hl ðxÞ and hl ðxÞ are the spherical Bessel function, Neumann function, and Hankel functions of the first and the second kinds. A1, A2, A3, B1, B2 and B3 are normalized constants. The boundary conditions for these functions are [26].
Rnl;1 ðR1 Þ ¼ Rnl;2 ðR1 Þ; Rnl;2 ðR2 Þ ¼ Rnl;3 ðR2 Þ; Rnl;3 ðR3 Þ ¼ 0;
(17)
1 dRnl;1 1 dRnl;2 ¼ ; m1 dr r¼R1 m2 dr r¼R1
(18)
ZR1
r 2 R*nl;1 ðrÞRnl;1 ðrÞdr
ZR2 þ
r2 R*nl;2 ðrÞRnl;2 ðrÞdr
R1
0
1 dRnl;2 1 dRnl;3 ¼ ; m2 dr r¼R2 m3 dr r¼R2 ZR3 þ
r 2 R*nl;3 ðrÞRnl;3 ðrÞdr ¼ 1:
(19)
R2
Because the wave function is limited when r/0, B1 ¼ 0. It is obvious that A1, A2, A3, B2, B3 and E0 can be obtained by solving the Eqs. (17)e(19). It is necessary to explain that l1(l2) and h1(h2) are variational parameters for the ground (excited) state to indicate the degree of spatial correlation between electron and impurity and the influence of the electric field, respectively. Then, the ground and excited state energies of the impurity state can be obtained by minimizing the expectation energies with respect to l1, h1 and l2, h2
〈4ðrÞjHj4ðrÞ〉 ; l1 ;h1 〈4ðrÞj4ðrÞ〉
(20)
〈4ðrÞjHj4ðrÞ〉 : 〈4ðrÞj4ðrÞ〉
(21)
E1 ¼ min E2 ¼ min l2 ;h2
2.2. Optical properties of the system In this work, an optical transition between the ground state and first excited state has been considered. The linear and third-order nonlinear ACs and RICs of the impurity state in a MSEQD can be obtained by a density matrix approach and a perturbation expansion method. The linear AC að1Þ ðuÞ and the third-order nonlinear AC að3Þ ðu; IÞ are calculated by Refs. [7,16,17,27].
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Fig. 2. Probability of finding the electron in ground and first excited states without electric field in various regions as a function of the core radius R1 for b ¼ 0 and Ww ¼ Bw ¼ 2 nm (a), or of the barrier shell width Bw for b ¼ 0, Ww ¼ 2 nm, and R1 ¼ 3 nm (b), or of the well width Ww for b ¼ 0, Bw ¼ 2 nm, and R1 ¼ 3 nm (c), or of the ellipticity constant b for R1 ¼ 3 nm and Ww ¼ Bw ¼ 2 nm (d). The Al content is chosen as x ¼ 0.3.
að1Þ ðuÞ ¼ u
ð3Þ
rffiffiffiffiffi
m
jM21 j2 sV ZG12
εR ðE21 ZuÞ2 þ ðZG12 Þ2
a ðu; IÞ ¼ u
rffiffiffiffiffi
m
εR
I 2ε0 nr c
h
;
(22) 2
jM21 j2 sV ZG12
i2 44jM21 j
2
ðE21 ZuÞ þ ðZG12 Þ i3 2 4E Zu þ Z2 u2 G2 jM22 M11 j2 3E21 21 12 5; 2 þ ðZG Þ2 E21 12 2
2
h
(23)
the total AC is defined as
aðuÞ ¼ að1Þ ðuÞ þ að3Þ ðu; IÞ:
(24)
The linear and the third-order nonlinear RICs are given by Refs. [7,17].
Dnð1Þ ðuÞ nr
¼
" # 1 E21 Zu 2 s ; jM21 j V 2n2r ε0 ðE21 ZuÞ2 þ ðZG12 Þ2
(25)
L. Shi, Z.W. Yan / Superlattices and Microstructures 94 (2016) 204e214
Dnð3Þ ðu; IÞ nr
209
h h ðM22 M11 Þ2 n 2 ðE21 ZuÞ E21 ðE21 ZuÞ i2 4ðE21 ZuÞjM21 j 2 2 E21 þ ðZG12 Þ ðE21 ZuÞ2 þ ðZG12 Þ2 i oi ðZG12 Þ2 ðZG12 Þ2 ð2E21 ZuÞ
mc ¼ 3 jM21 j2 h 4nr ε0
sV I
(26) the total RIC is defined as
Dnðu; IÞ nr
¼
Dnð1Þ ðu; IÞ nr
þ
Dnð3Þ ðu; IÞ nr
;
(27)
where m is the magnetic susceptibility, sV is the electron density, nr is the refractive index of dot material, ħu is the incident photon energy, εR ¼ n2r ε0 is the real part of the permittivity, I is the incident light intensity, G12 is the relaxation rate for the ground and excited states, E21 ¼ E2E1 is the energy difference between two different electronic states, and M21 is the transition matrix element between the initial and final states. Here, we have chosen a polarized electromagnetic radiation in the z-direction, and hence the transition matrix element is defined as
M21 ¼ j〈42 jezj41 〉j:
(28)
3. Results and discussion In this study, the numerical calculations have been carried out for a GaAs/AlxGa1xAs/GaAs MSEQD system by using the following material parameters: sV ¼ 3.0 1016 cm3, nr ¼ 3.2, T12 ¼ 0.2 ps, G12 ¼ 1/T12 [27]. The effective mass and static dielectric are: m1 ¼ 0.067m0, m2 ¼ (0.067 þ 0.083)m0 (m0 is the mass of a free electron), and ε1 ¼ ε3 ¼ 13.18, ε2 ¼ 13.18 e 3.12x [28]. The band offset for the GaAs/AlxGa1xAs is given as V0 ¼ 0.7(1247x) meV [28]. In order to understand the problem more clearly in the next discussion, firstly the probability of finding the electron in the ground and first excited states in different regions are shown in Fig. 2. The barrier shell, well width and core radius for each figure are chosen to allow the electron tunneling between each region. Fig. 2(a) shows that for the very small core radius, the electron in the ground state is mainly confined in the core region, but the electron in the excited state is mainly confined in the
Fig. 3. Transition energy as a function of electric field for various R1 values with b ¼ 0 and Ww ¼ Bw ¼ 2 nm, or for various Bw values with b ¼ 0, Ww ¼ 2 nm, and R1 ¼ 3 nm, or for various Ww values with b ¼ 0, Bw ¼ 2 nm, and R1 ¼ 3 nm, or for various ellipticity constant b with R1 ¼ 3 nm, Ww ¼ Bw ¼ 2 nm. The Al content is chosen as x ¼ 0.3.
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Fig. 4. The linear (dotted lines), third-order nonlinear (dashed lines) and total (solid lines) ACs (a) and RICs (b) with and without electric field as a function of photon energy ħu for various R1 and b values with Ww ¼ Bw ¼ 2 nm, x ¼ 0.3 and I ¼ 0.3 MW/cm2. Red, dark and blue lines correspond to b ¼ 0.2, 0, 0.2, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
barrier shell region. As expected, as the core radius increases the penetration of the electron wave function into shell and well regions is weakened due to the impurity, therefore the electron in the excited state is also mainly confined in the core region. Fig. 2(b) shows that the electron in the ground state is mainly confined in the core region, and the variation of barrier shell thickness has little effect on the ground state. By contrast, the electron in the excited state is mainly confined in the shell region, and this confinement is obviously enhanced as the shell width increases. Fig. 2(c) shows that as the well width increases the penetration of the electron wave functions into well region is greatly increased but that into core and shell regions is decreased, and finally the electron in ground and excited states is mainly confined in the well region. Fig. 2(d) shows that the variation of the b value has little effect on the probability. Because the electric field and shape have an important influence on the transition energies, Fig. 3 shows the transition energies as a function of electric field under different conditions. It is found that the increasing electric field decreases the transition energies, and this behavior is more pronounced for the larger core radius or shell thickness or well width. In particular, we found that the influence of electric field is more pronounced for larger b values, which means that the prolate case is more sensitive to the electron field compared with the oblate case. This is because the prolate case expands in the zdirection size of the MSEQD, which is just the direction of the electric field, and therefore the influence of the electric field is enhanced. The ACs and RICs with and without the electric field as a function of the photon energy for different core radii and b values are plotted in Fig. 4. It is seen that the increasing core radius and/or electric field and/or b value red shifts the peak positions of linear, nonlinear and total ACs and RICs, and the red shift caused by the electric field is more pronounced for the larger core radius, this is because the increasing core radius and/or electric field and/or b value decreases the transition energy, and the
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211
Fig. 5. The linear (dotted lines), third-order nonlinear (dashed lines) and total (solid lines) ACs (a) and RICs (b) with and without electric field as a function of photon energy ħu for various Bw and b values with R1 ¼ 3 nm, Ww ¼ 2 nm, x ¼ 0.3 and I ¼ 0.3 MW/cm2. Red, dark and blue lines correspond to b ¼ 0.2, 0, 0.2, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
influence of electric field is more pronounced for the larger core radius as shown in Fig. 3. However, as expected, the red shift caused by b value is more pronounced for the smaller core radius, by reason of the effect of shape will become more and more weak as the size of MSEQD such as core radius, shell thickness and well width is increased. In addition, it is found that the peak intensities of nonlinear ACs, linear and nonlinear RICs increase with the increasing core radius, but this behavior is insensitive to the linear ACs. It can be explained that because the increasing core radius suppresses the tunneling effect and therefore the electron is mainly confined in core region as can be seen in Fig. 2(a), the overlapping of the wave functions of the initial and final states is increased, and the peak intensities of nonlinear ACs, linear and nonlinear RICs are increased. However, the peak intensity of linear ACs also depends on the transition energy E21 besides the overlapping of the wave functions. In contrast to the increasing overlapping of the wave functions, the transition energy experiences a decrease with increasing core radius as shown in Fig. 3. The competition effect between these two factors causes the linear ACs is insensitive to the variation of core radius. Fig. 5 shows the ACs and RICs with and without electric field as a function of the photon energy for different shell thickness. It shows that the increasing shell thickness and/or electric field and/or b value red shifts the peak positions of linear, nonlinear and total ACs and RICs, and the red shift caused by the electric field is more pronounced for the larger shell thickness. It can be also explained by Fig. 3 which shows that the increasing shell thickness and/or electric field and/or b value decreases the transition energy, and the influence of electric field on the transition energy is more pronounced for the larger shell thickness. Similar to Fig. 4, the red shift caused by b value is more pronounced for the smaller shell thickness. In contrast to Fig. 4, it is found that the increasing shell thickness decreases the peak intensities of linear and nonlinear ACs, but almost no effect on RICs. It can be explained that the electron in the ground state is mainly confined in the core region, but the increasing shell thickness obviously facilitates the electron in the excited state confined in the shell region as shown in Fig. 2(b).
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Fig. 6. The linear (dotted lines), third-order nonlinear (dashed lines) and total (solid lines) ACs (a) and RICs (b) with and without electric field as a function of photon energy ħu for various Ww and b values with R1 ¼ 3 nm, Bw ¼ 2 nm, x ¼ 0.3 and I ¼ 0.2 MW/cm2. Red, dark and blue lines correspond to b ¼ 0.2, 0, 0.2, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Therefore, the overlapping of the wave functions of the initial and final states is decreased, and their peak intensities are decreased. However, the variation of the overlapping of the wave functions is too small to obviously change the peak intensities of RICs, and therefore lead to the results as shown. The ACs and RICs with and without electric field as a function of the photon energy for different well widths are plotted in Fig. 6. It shows that the increasing well width and/or electric field and/or b value red shifts the peak positions of linear, nonlinear and total ACs and RICs, and the red shift caused by the electric field is more pronounced for the larger well width, but that caused by b value is more pronounced for the smaller well width. It can be also explained by Fig. 3 which shows that the increasing well width and/or electric field and/or b value decreases the transition energy, and the influence of electric field on the transition energy is more pronounced for the larger well width. In addition, it is found that the increasing well width obviously increases the peak intensities of linear and nonlinear RICs and nonlinear ACs, but it has little effect on the linear ACs. The reason can be found in Fig. 2(c), which shows that the electron in ground and excited states is more and more confined in the well region with the increasing well width, and therefore the overlapping of the wave functions of the initial and final states is increased, which opposites to the variation of transition energy. The competition effect between these two factors causes the linear ACs is insensitive to the variation of well width. In particular, it can be seen that the peak of total AC for Ww ¼ 6 nm under the electric field obviously splits into two peaks, and the split is enhanced with increasing b. Finally, the ACs and RICs with and without electric field as a function of the photon energy for different Al contents are presented in Fig. 7. It shows that the increasing Al content blue shifts the peak positions of linear, nonlinear and total ACs and RICs, but the increasing electric field and/or b value red shift them. This is because the transition energy is increased by the increasing Al content due to the quantum size effect, but reduced by the electric field. In addition, it is found that the peak intensities of linear ACs almost no change with the increasing Al content, but the linear and nonlinear RICs and nonlinear ACs
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Fig. 7. The linear (dotted lines), third-order nonlinear (dashed lines) and total (solid lines) ACs (a) and RICs (b) with and without electric field as a function of photon energy ħu for various x and b values with R1 ¼ 6 nm, Ww ¼ Bw ¼ 2 nm, and I ¼ 0.3 MW/cm2.
decrease monotonically with it. It is because the increasing Al content increases the energy level difference due to the quantum size effect, the overlapping of the wave functions of the initial and final states is decreased, which counteracts the increasing transition energy. 4. Conclusions In summary, the linear and nonlinear optical properties of a GaAs/AlxGa1xAs/GaAs MSEQD under the influence of an external field are studied. The results indicate that the electric field red shifts the peak positions of the ACs and RICs, and its
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influence is more pronounced for the large regions, i.e. core, shell and well ones. By contrast, the increasing Al content blue shifts the peak positions. In addition, we observe that the shape of MSEQD also has an important influence on the ACs and RICs. Compared with the spherical case (b ¼ 0), the prolate case (b > 0) red shifts the peak positions of the ACs and RICs, but the oblate case (b < 0) blue shifts them. This study provides an understanding of the optical properties of MSQD structures and how the optical properties are affected by the electric field. We hope that the present theory can stimulate further studies related to the optical properties of MSEQDs, and help experimenters use electric field to modulate the optical properties of MSEQDs. Acknowledgments This work was supported by the National Natural Science Foundation of PR China (Project No. 11364028), the Major Projects of Natural Science Foundation of Inner Mongolia (Project 2013ZD02), the Doctoral Staring up Foundation of Inner Mongolia Agricultural University (Project BJ2013B-2), the Grassland talent project. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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