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Magnetic field and hydrostatic pressure effects on electron Raman scattering in anisotropic quantum dots
Accepted Manuscript Magnetic field and hydrostatic pressure effects on electron Raman scattering in anisotropic quantum dots Wenfang Xie PII: DOI: Reference:
S0301-0104(13)00269-3 http://dx.doi.org/10.1016/j.chemphys.2013.06.013 CHEMPH 8900
To appear in:
Chemical Physics
Received Date: Accepted Date:
2 May 2013 18 June 2013
Please cite this article as: W. Xie, Magnetic field and hydrostatic pressure effects on electron Raman scattering in anisotropic quantum dots, Chemical Physics (2013), doi: http://dx.doi.org/10.1016/j.chemphys.2013.06.013
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Magnetic field and hydrostatic pressure effects on electron Raman scattering in anisotropic quantum dots
Wenfang Xie School of Physics and Electronic Engineering, Guangzhou University Guangzhou 510006, P. R. China
Abstract
We have investigated the electron Raman scattering process of a two-dimensional anisotropic quantum dot. With typical semiconducting GaAs based materials, the differential cross-section has been examined on the basis of the computed energies and wave functions. We also studied the effects of external magnetic field and hydrostatic pressure on the Raman scattering in anisotropic quantum dots. The results show that electron Raman scattering in anisotropic QDs is strongly affected by the degree of anisotropy, dot size, applied magnetic field and hydrostatic pressure. It is possible to control the frequency shift and the peak intensity of the Raman spectrum in anisotropic QDs by varying these factors.
Keywords: Quantum dot; Anisotropy; Raman scattering
Tel number: (8620)83551467 E-mail address: [email protected]
1
1 Introduction Recent advances in modern nanotechnology enable the fabricate quantum dots (QDs) of different shapes [1, 2]. Optical properties of QDs can be controlled by modifying the dot size, the geometrical shape, and the confining potential. In addition, it is particularly important to investigate the optical properties of QDs because the study of intersubband transitions yields important information about the energy spectrum (and other related quantities). It is now well known that an electron, which is confined into dimensions of a few tens of nanometers provides strong blueshift of the photoluminescence features from that in the original bulk material, a clear consequence of quantum confinement. Optical methods are convenient experimental tools for studying the properties of QDs. In particular, the electronic structure of QDs can be studied through the Raman scattering processes considering different polarizations of incident and emitted radiation [3 − 5]. The calculation of the differential cross-section (DCS) of Raman scattering remains a rather interesting and fundamental issue to achieve a better understanding of the electronic and optical properties in semiconductor QDs. Recently, some experimental [6 − 10] and theoretical [11 − 19] studies have been focused on the Raman scattering and photoluminescence in QDs. They found that the Raman scattering in QDs strongly depends on the dot size and the strength of the external field. This is because most of the physical properties are implicitly contained in the wave function. Any change in the wave function, due to effect of the external factors, changes the physical properties of the system. Hence external factors such as temperature, electric and magnetic fields, and pressure can change 2
the optical properties of QDs. It is well known that high hydrostatic pressure is a thermodynamic variable for the solid state that can provide important information to enable the understanding of the electronic properties on QDs. Hence, hydrostatic pressure is a powerful tool to investigate and control the electronic-related optical properties of low-dimensional semiconductor systems. Recently, the effect of hydrostatic pressure on the electronic and optical properties of QDs has also attracted enormous interest because they have the potential for device applications [20 − 24]. However, most previous studies have focused on some symmetric confinement potentials, for anisotropy significantly complicates the mathematical analysis. This is unfortunate, for most real systems are not isotropic, and it is therefore important to understand how anisotropy affects. Recently, there have been some experimental studies on elliptical QDs [25]. Also, the confinement potential of QDs studied by McEuen et al. was found to be anisotropic [26, 27]. Geyler and co-worker studied the hybrid resonances of the optical absorption in a three-dimensional anisotropic QD [28]. Very recently, Khordad and co-worker investigated the optical and electronic properties of anisotropic parabolic quantum disks in the presence of tilted magnetic fields [29]. They found that the total absorption coefficients increase as the tilt angle of magnetic field and anisotropy increase. Obviously, for most real systems are not isotropic, and it is therefore important to understand how the anisotropy affects QDs. In present work, we will focus on studying the electron Raman scattering of a two-dimensional anisotropic parabolic QD under external magnetic filed. We will examine the DCS for the different anisotropy cases. Furthermore, we have investigated the influence of external magnetic fields and hydrostatic pressure on the DCS 3
of the electron Raman scattering of a two-dimensional anisotropic QD. 2 Method of calculation Let us consider an electron confined in lateral anisotropic parabolic potential in the presence of a perpendicular magnetic field. We assume that the system is in the temperature T = 0K. So, the interaction between the phonon and the electron is eliminated and does not have any effect on optical properties of semiconductor QDs. Within the effective-mass approximation, the Hamiltonian of this system can be written as H=
1 e~ 2 1 (~p + A) + me (P, T )(Ω2x x2 + Ω2y y 2 ), 2me (P, T ) c 2
(1)
where me (P, T ) is the electron effective masse which is dependent of the hydrostatic pressure (P ) and temperature (T ), p~ is the momentum vector of the electron, e is the electron charge, and c is the speed of light. The two-dimensional confinement is modeled via harmonic potentials with two different frequencies Ωx and Ωy [29 − 31], which yield the elliptical cross sections of the dots with axes ratio given by Ly /Lx = q
Ωx /Ωy , where Li =
q
h ¯ /me (p, T )Ωi (i = x, y). Hence, the ratio Ωx /Ωy characterizes
the degree of anisotropy of the QD. The effects of hydrostatic pressure and temperature are incorporated via the variation of the electron and hole effective masses and the static dielectric constant up on both variables. For the electron effective masses, we have [32] me (P, T ) =
1+
m0 , + ( E Γ (P,T1 )+∆0 )]
2 EPΓ [( E Γ (P,T ) ) g
(2)
g
where m0 is the single electron bare mass, EPΓ = 7.51eV is an energy related to the momentum matrix element, and ∆0 = 0.341eV is the spin–orbit splitting. EgΓ (P, T ) 4
is the Γ related energy gap for the GaAs QD [33 − 35], by EgΓ (P, T ) = EgΓ (0, T ) + aP + bP 2 ,
(3)
where a = 1.07 × 10−2 eV /kbar, b = −3.77 × 10−5 eV /kbar2 , and EgΓ (0, T )
5.405 × 10−4 T 2 = [1.519 − ]eV. T + 204
(4)
The hydrostatic pressure and temperature are given in kbar and K (Kelvin) degrees, respectively. It is obvious that the electron effective mass in GaAs QDs is an increasing function of the hydrostatic pressure and a decreasing function of the ~ = (B/2)(−y, x, 0), temperature. With the symmetry gauge for the magnetic field A the Hamiltonian is then
H=
1 [p2 + Ω21 x2 + p2y + Ω22 y 2 + me (P, T )ωc (ypx − xpy )], 2me (P, T ) x
(5)
where Ω21 = me (P, T )2 (Ω2x + ωc2 /4), Ω22 = me (P, T )2 (Ω2y + ωc2 /4).
(6)
Here ωc = eB/me (P, T )c is the cyclotron frequency of the electron. We make the following transformations [30]: x = q1 cos χ −
χ2 p2 sin χ, χ
y = q2 cos χ −
χ2 p1 sin χ, χ
px = p1 cos χ +
χ1 q2 sin χ, χ
py = p2 cos χ +
χ1 q1 sin χ. χ
5
(7)
If χ1 χ2 = χ2 , these operators satisfy the commutation relations [qi , pj ] = −i¯ hδij , [qi , qj ] = 0, and [pi , pj ] = 0. Substituting (7) into (5) we get the new Hamiltonian which is diagonal
H=
1 (α12 p21 + α22 p22 + β12 q12 + β22 q22 ), 2me (P, T )
(8)
where αi (i = 1, 2) and βi are defined as α12 =
Ω21 + 3Ω22 + Ω23 , 2(Ω21 + Ω22 )
α22 =
3Ω21 + Ω22 − Ω23 , 2(Ω21 + Ω22 )
β12 = (3Ω21 + Ω22 + Ω23 )/4, β22 = (Ω21 + 3Ω22 + Ω23 )/4.
(9)
In the above equations, Ω3 is defined as Ω23 = [(Ω21 − Ω22 )2 + 2me (P, T )2 ωc2 (Ω21 + Ω22 )]1/2 .
(10)
Hence, the energy eigenvalues in anisotropic QDs are given by 1 1 En1 n2 = (n1 + )¯ hω1 + (n2 + )¯ hω2 , 2 2
n1 , n2 , = 0, 1, 2, . . . ,
(11)
where ω1 = α1 β1 /me (P, T ) and ω2 = α2 β2 /me (P, T ). And the corresponding wave functions in the new phase coordinates have the form
Φn1 n2 (q1 , q2 ) = Φn1 (q1 )Φn2 (q2 ), where Φn (q) is the wave function of one-dimensional oscillator.
6
(12)
The interaction of an electromagnetic wave in the far infrared range with electrons confined in a QD can be described in a dipole approximation. For an electron, the interaction Hamiltonian is given by ~ 0 · ~r exp(iωt), Hinter = eE
(13)
where E0 is the amplitude of oscillations of the electric field. The Raman cross section can be given by the perturbation theory. The DCS for electron Raman scattering in a volume per unit solid angle for incoming light of frequency ωl , and scattered light of frequency ωs , is given by by Ref. [35] d2 σ V 2 ω 2 η(ωs ) = 3 s4 W (ωs , ~es ), dΩdωs 8π c η(ωl )
(14)
where η(ω) is the refraction index as a function of the radiation frequency, ~es is the unit polarization vector for the emitted secondary radiation, c is the light velocity in vacuum and W (ωs , ~es ) is the transition rate for the emission of secondary radiation (with frequency ωs and polarization ~es ), which is calculated according to Fermi’s golden rule: W (ωs , ~es ) =
2π X |Mf i |2 δ(Ef − Ei ). h ¯ f
(15)
In Eq. (14), Mf i and δ(Ef − Ei ) are defined by Mf i =
X hf |Hs |aiha|Hl |ii a
Ei − Ea + iΓa
,
(16)
and δ(Ef − Ei ) =
Γf . π[(Ef − Ei )2 + Γ2f ]
(17)
Here, |ii, |ai and |f i denote initial, intermediate and final states of the system with their corresponding energies Ei , Ea and Ef , respectively. These energies are 7
determinable by using the variational method. And Γf is the life-time width. The operator Hl is of the form
|e| Hl = m0
s
2π¯ h ~el · p~, V ωl
p~ = −i¯ h∇.
(18)
This operator describes the interaction with the incident radiation field in the dipole approximation. The Hamiltonian of the interaction with the secondary radiation field is given by |e| Hs = me (P, T )
s
2π¯ h ~es · p~, V ωs
p~ = −i¯ h∇.
(19)
Here ~el is the unit polarization vector for the incident radiation. This Hamiltonian describes the photon emission by the electron after transitions between conduction subbands of the system. The selection rules between the energy levels can be calculated from the dipole matrix elements and are as follows: polarization along the x or y axis: ∆n1 = ±1, ∆n2 = 0, and the y axis: ∆n1 = 0, ∆n2 = ±1. In this work, the DCS of Raman scattering for a three-level system is calculated for a QD using equation (13). We choose the backscattering configuration, where the incident radiation wave vector is parallel to the z axis, i.e., ki ||z, and the scattered radiation wave vector parallel to the z axis, i.e., kf ||z, with both the polarizations of the incident and scattered wave vectors parallel to the x axis and y axis, i.e., Z(X, Y )Z. In the initial state we have an electron in the Ψ01 state and an incident photon of energy h ¯ ωl with polarization along the x axis. From the Ψ01 state the electron out a transition to the intermediate state Ψ11 with energy Ea . From the intermediate state the electron undergoes a transition toward the Ψ10 state, emitting the laser
8
radiation as secondary radiation of energy h ¯ ωs with polarization along the y axis. 3 Results and discussions In this section we will calculate the influence of an external magnetic field, hydrostatic pressure, temperature and the dot parameters on the DCS of electron Raman scattering in a two-dimensional typical GaAs anisotropic QD. The lifetimes of the final and intermediate states are Γf = Γa = 1.0meV . All calculations of electron Raman scattering are performed in the scattering configuration Z(X, Y )Z. Fig. 1 shows the variation of the DCS as a function of the diffusion photon energy for three different values of the degree of anisotropy, i.e., Ωx /Ωy = 2.25, 4.0, and 9.0 (Ly /Lx = 1.5, 2.0, and 3.0), respectively. The QD size is defined as the confinement characteristic length R =
q
Lx Ly . In Fig. 1, the dot size R is set to be 6.0nm. Here,
hydrostatic pressure and magnetic field are set to be P = 20kbar and B = 10T , respectively. In Fig. 1, the DCS values of Ωx /Ωy = 4.0 and 9.0 should multiply 0.2 and 0.01 times, respectively. From this figure we can find that the degree effect of anisotropy of the QD on Raman scattering is significant. We observe that the rapid decrease in resonance peak of the DCS results from the increasing the anisotropy degree of QDs. And the resonance peak of the DCS moves towards lower energies (red shift) with increasing the ratio Ωx /Ωy . These results can be easily explained, when the ratio Ωx /Ωy increases, the confinement of the y direction will decrease and the confinement of the x direction will increase. This leads to that the energy difference Ef a between the Ψa and Ψf states decreases and the energy difference Eai between the Ψi and Ψa states increases. Hence, the resonant peaks of all DCS suffer an obvious red-shift and the peak intensities rapid decrease with increasing the ratio 9
Ωx /Ωy . It is possible to manage the resonance frequency and the amplitude of the DCS by controlling the anisotropy degree of QDs. In order to investigate the influence of geometrical size of QDs, we plot, in Fig. 2, the DCS as a function of the diffusion photon energy for three different dot sizes. In Fig. 2, the degree of anisotropy, hydrostatic pressure, and magnetic field are set to be Ωx /Ωy = 2.25, P = 20kbar, and B = 10T , respectively. The plots indicate that all peak positions of DCS shift to lower energies (red shift) with increasing R. This result is in good agreement with those obtained in spherical QDs [36, 37]. Figure 2 also shows that, as R is increased, the peak intensities of the DCS decrease. Similar behavior is also observed in Ref. [37]. Obviously, the confinements of the x and the y directions decrease with increasing dot size R so that the energy differences Ef a and Eia decrease. The resonant peaks of all DCS suffer an obvious red-shift and the peak intensities decrease with increasing R. Therefore, we can say that the smaller dot size should used in order to obtain the considerable changes in the Raman scattering of anisotropic QDs. Fig. 3 shows the effect of external magnetic field on the Raman scattering in a two-dimensional anisotropic QD. We plot the DCS as a function of the diffusion photon energy for three different strengths of magnetic field. In Fig. 3, the degree of anisotropy, hydrostatic pressure and dot size are set to be Ωx /Ωy = 2.25, P = 20kbar and R = 6.0nm, respectively. From this figure the magnetic field-induced effect on the Raman scattering of anisotropic QDs is clear. It is readily seen that the resonant peaks of all DCS suffer an obvious red-shift and the peak intensities decrease with increasing B. This result is in good agreement with that obtained in a cylindrical 10
QD [15]. The physical origin of these results is that the energy difference Ef a decreases and the energy difference Eai states increases with increasing B. This phenomenon tells us that by varying the strength of external magnetic field, it is possible to control the frequency shift and the peak intensity in the Raman spectrum of anisotropic QDs. It is well known that application of hydrostatic pressure results in a modification of the physical properties [38, 39]. This mainly is due to deformation of the interatomic bonds [40]. Furthermore, in order to investigate the influence of the hydrostatic pressure on the Raman scattering in anisotropic QDs, we plot, in Fig. 4, the DCS as a function of the diffusion photon energy for three different hydrostatic pressure values, i.e., P = 0, 20 and 40kbar, respectively. Here, the degree of anisotropy, dot size and magnetic field strength are set to be Ωx /Ωy = 2.25, R = 6.0nm, and B = 10T , respectively. As we can see from Fig. 4 that as a result of the increase in the pressure, the magnitude of the DCS decreases and the Raman spectrum shows a red shift. This is obtained due to the increase of the electron effective mass with increasing hydrostatic pressure. Therefore, an increase in the hydrostatic pressure leads to the more confinement of electrons in QDs and thus weakens the DCS of Raman scattering. The obtained results give a new degree of freedom in device applications, such as photo-detectors, electro-optical modulators, and all optical switches. In conclusion, we have investigated the effects of hydrostatic pressure, and external magnetic field on electron Raman scattering in a two-dimensional anisotropic QD. The study has considered the simultaneous effects of external magnetic field 11
and hydrostatic pressure for several values of the degree of anisotropy of QDs. With typical semiconductor GaAs based materials, the DCS as a function of the diffusion photon energy have been obtained. The results show that the electron Raman scattering in anisotropic QDs is strongly affected by the degree of anisotropy, dot size, and applied magnetic field strength and hydrostatic pressure of QDs. We found that the resonant peak of the DCS of Raman scattering in anisotropic QDs can be red or blue shifted by external probes such as applied magnetic field and hydrostatic pressure. The larger change of the Raman spectrum will be obtained by controlling the anisotropy degree of QDs. The dependence of DCS on size of the QD, the magnetic field and the hydrostatic pressure could be used for spectroscopic characterization of low-dimensional semiconductor systems. Acknowledgment: This work is supported by the National Natural Science Foundation of China under Grant No. 11074055.
12
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Figure captions: Figure 1 Dependences of the DCS of Raman scattering in a two-dimensional anisotropic QD as a function of photon energy for the three different Ωx /Ωy values with P = 20kbar, R = 6.0nm and B = 10T . Figure 2 Dependences of the DCS of Raman scattering in a two-dimensional anisotropic QD as a function of photon energy for the three different R values with P = 20kbar, Ωx /Ωy = 2.25 and B = 10T . Figure 3 Dependences of the DCS of Raman scattering in a two-dimensional anisotropic QD as a function of photon energy for the three different B values with P = 20kbar, Ωx /Ωy = 2.25 and R = 6.0nm. Figure 4 Dependences of the DCS of Raman scattering in a two-dimensional anisotropic QD as a function of photon energy for the three different P values with B = 10T , Ωx /Ωy = 2.25 and R = 6.0nm.
16
Ωx/Ωy=2.25
Figure 1
Ωx/Ωy=3.0
P=20.0kbar,B=10.0T,R=6.0nm
Ωx/Ωy=9.0
DCS (arb.units)
0.6
0.4
×0.2 0.2
×0.01
0.0
0
10
20
30
hωs (meV)
40
50
Figure 2
R=6.0nm R=7.0nm R=8.0nm
P=20.0kbar,B=10.0T,Ωx/Ωy=2.25
DCS (arb.units)
0.6
0.4
0.2
0.0
0
10
20
30
hωs (meV)
40
50
B=5.0T B=10.0T B=15.0T
Figure 3 1.0
P=20.0kbar,R=6.0nm,Ωx/Ωy=2.25
DCS (arb.units)
0.8
0.6
0.4
0.2
0.0
0
10
20
30
hωs (mev)
40
50
Figure 4
P=0kbar P=20.0kbar P=40.0kbar
B=10.0T,R=6.0nm,Ωx/Ωy=2.25
DCS (arb.units)
0.8
0.6
0.4
0.2
0.0
0
10
20
30
hωs (meV)
40
50
Highlights：
1. Electron Raman scattering process of a two-dimensional anisotropic quantum dot is investigated. 2. The larger peak intensity will be obtained by controlling the anisotropy degree. 3. It is possible to control the frequency shift by external probes such as magnetic field and hydrostatic pressure.
Ωx/Ωy=2.25 Ωx/Ωy=3.0 Ωx/Ωy=9.0
DCS (arb.units)
0.6
0.4
×0.2 0.2
×0.01
0.0
0
10
20
30
hωs (meV)
40
50
S0301-0104(13)00269-3 http://dx.doi.org/10.1016/j.chemphys.2013.06.013 CHEMPH 8900
To appear in:
Chemical Physics
Received Date: Accepted Date:
2 May 2013 18 June 2013
Please cite this article as: W. Xie, Magnetic field and hydrostatic pressure effects on electron Raman scattering in anisotropic quantum dots, Chemical Physics (2013), doi: http://dx.doi.org/10.1016/j.chemphys.2013.06.013
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Magnetic field and hydrostatic pressure effects on electron Raman scattering in anisotropic quantum dots
Wenfang Xie School of Physics and Electronic Engineering, Guangzhou University Guangzhou 510006, P. R. China
Abstract
We have investigated the electron Raman scattering process of a two-dimensional anisotropic quantum dot. With typical semiconducting GaAs based materials, the differential cross-section has been examined on the basis of the computed energies and wave functions. We also studied the effects of external magnetic field and hydrostatic pressure on the Raman scattering in anisotropic quantum dots. The results show that electron Raman scattering in anisotropic QDs is strongly affected by the degree of anisotropy, dot size, applied magnetic field and hydrostatic pressure. It is possible to control the frequency shift and the peak intensity of the Raman spectrum in anisotropic QDs by varying these factors.
Keywords: Quantum dot; Anisotropy; Raman scattering
Tel number: (8620)83551467 E-mail address: [email protected]
1
1 Introduction Recent advances in modern nanotechnology enable the fabricate quantum dots (QDs) of different shapes [1, 2]. Optical properties of QDs can be controlled by modifying the dot size, the geometrical shape, and the confining potential. In addition, it is particularly important to investigate the optical properties of QDs because the study of intersubband transitions yields important information about the energy spectrum (and other related quantities). It is now well known that an electron, which is confined into dimensions of a few tens of nanometers provides strong blueshift of the photoluminescence features from that in the original bulk material, a clear consequence of quantum confinement. Optical methods are convenient experimental tools for studying the properties of QDs. In particular, the electronic structure of QDs can be studied through the Raman scattering processes considering different polarizations of incident and emitted radiation [3 − 5]. The calculation of the differential cross-section (DCS) of Raman scattering remains a rather interesting and fundamental issue to achieve a better understanding of the electronic and optical properties in semiconductor QDs. Recently, some experimental [6 − 10] and theoretical [11 − 19] studies have been focused on the Raman scattering and photoluminescence in QDs. They found that the Raman scattering in QDs strongly depends on the dot size and the strength of the external field. This is because most of the physical properties are implicitly contained in the wave function. Any change in the wave function, due to effect of the external factors, changes the physical properties of the system. Hence external factors such as temperature, electric and magnetic fields, and pressure can change 2
the optical properties of QDs. It is well known that high hydrostatic pressure is a thermodynamic variable for the solid state that can provide important information to enable the understanding of the electronic properties on QDs. Hence, hydrostatic pressure is a powerful tool to investigate and control the electronic-related optical properties of low-dimensional semiconductor systems. Recently, the effect of hydrostatic pressure on the electronic and optical properties of QDs has also attracted enormous interest because they have the potential for device applications [20 − 24]. However, most previous studies have focused on some symmetric confinement potentials, for anisotropy significantly complicates the mathematical analysis. This is unfortunate, for most real systems are not isotropic, and it is therefore important to understand how anisotropy affects. Recently, there have been some experimental studies on elliptical QDs [25]. Also, the confinement potential of QDs studied by McEuen et al. was found to be anisotropic [26, 27]. Geyler and co-worker studied the hybrid resonances of the optical absorption in a three-dimensional anisotropic QD [28]. Very recently, Khordad and co-worker investigated the optical and electronic properties of anisotropic parabolic quantum disks in the presence of tilted magnetic fields [29]. They found that the total absorption coefficients increase as the tilt angle of magnetic field and anisotropy increase. Obviously, for most real systems are not isotropic, and it is therefore important to understand how the anisotropy affects QDs. In present work, we will focus on studying the electron Raman scattering of a two-dimensional anisotropic parabolic QD under external magnetic filed. We will examine the DCS for the different anisotropy cases. Furthermore, we have investigated the influence of external magnetic fields and hydrostatic pressure on the DCS 3
of the electron Raman scattering of a two-dimensional anisotropic QD. 2 Method of calculation Let us consider an electron confined in lateral anisotropic parabolic potential in the presence of a perpendicular magnetic field. We assume that the system is in the temperature T = 0K. So, the interaction between the phonon and the electron is eliminated and does not have any effect on optical properties of semiconductor QDs. Within the effective-mass approximation, the Hamiltonian of this system can be written as H=
1 e~ 2 1 (~p + A) + me (P, T )(Ω2x x2 + Ω2y y 2 ), 2me (P, T ) c 2
(1)
where me (P, T ) is the electron effective masse which is dependent of the hydrostatic pressure (P ) and temperature (T ), p~ is the momentum vector of the electron, e is the electron charge, and c is the speed of light. The two-dimensional confinement is modeled via harmonic potentials with two different frequencies Ωx and Ωy [29 − 31], which yield the elliptical cross sections of the dots with axes ratio given by Ly /Lx = q
Ωx /Ωy , where Li =
q
h ¯ /me (p, T )Ωi (i = x, y). Hence, the ratio Ωx /Ωy characterizes
the degree of anisotropy of the QD. The effects of hydrostatic pressure and temperature are incorporated via the variation of the electron and hole effective masses and the static dielectric constant up on both variables. For the electron effective masses, we have [32] me (P, T ) =
1+
m0 , + ( E Γ (P,T1 )+∆0 )]
2 EPΓ [( E Γ (P,T ) ) g
(2)
g
where m0 is the single electron bare mass, EPΓ = 7.51eV is an energy related to the momentum matrix element, and ∆0 = 0.341eV is the spin–orbit splitting. EgΓ (P, T ) 4
is the Γ related energy gap for the GaAs QD [33 − 35], by EgΓ (P, T ) = EgΓ (0, T ) + aP + bP 2 ,
(3)
where a = 1.07 × 10−2 eV /kbar, b = −3.77 × 10−5 eV /kbar2 , and EgΓ (0, T )
5.405 × 10−4 T 2 = [1.519 − ]eV. T + 204
(4)
The hydrostatic pressure and temperature are given in kbar and K (Kelvin) degrees, respectively. It is obvious that the electron effective mass in GaAs QDs is an increasing function of the hydrostatic pressure and a decreasing function of the ~ = (B/2)(−y, x, 0), temperature. With the symmetry gauge for the magnetic field A the Hamiltonian is then
H=
1 [p2 + Ω21 x2 + p2y + Ω22 y 2 + me (P, T )ωc (ypx − xpy )], 2me (P, T ) x
(5)
where Ω21 = me (P, T )2 (Ω2x + ωc2 /4), Ω22 = me (P, T )2 (Ω2y + ωc2 /4).
(6)
Here ωc = eB/me (P, T )c is the cyclotron frequency of the electron. We make the following transformations [30]: x = q1 cos χ −
χ2 p2 sin χ, χ
y = q2 cos χ −
χ2 p1 sin χ, χ
px = p1 cos χ +
χ1 q2 sin χ, χ
py = p2 cos χ +
χ1 q1 sin χ. χ
5
(7)
If χ1 χ2 = χ2 , these operators satisfy the commutation relations [qi , pj ] = −i¯ hδij , [qi , qj ] = 0, and [pi , pj ] = 0. Substituting (7) into (5) we get the new Hamiltonian which is diagonal
H=
1 (α12 p21 + α22 p22 + β12 q12 + β22 q22 ), 2me (P, T )
(8)
where αi (i = 1, 2) and βi are defined as α12 =
Ω21 + 3Ω22 + Ω23 , 2(Ω21 + Ω22 )
α22 =
3Ω21 + Ω22 − Ω23 , 2(Ω21 + Ω22 )
β12 = (3Ω21 + Ω22 + Ω23 )/4, β22 = (Ω21 + 3Ω22 + Ω23 )/4.
(9)
In the above equations, Ω3 is defined as Ω23 = [(Ω21 − Ω22 )2 + 2me (P, T )2 ωc2 (Ω21 + Ω22 )]1/2 .
(10)
Hence, the energy eigenvalues in anisotropic QDs are given by 1 1 En1 n2 = (n1 + )¯ hω1 + (n2 + )¯ hω2 , 2 2
n1 , n2 , = 0, 1, 2, . . . ,
(11)
where ω1 = α1 β1 /me (P, T ) and ω2 = α2 β2 /me (P, T ). And the corresponding wave functions in the new phase coordinates have the form
Φn1 n2 (q1 , q2 ) = Φn1 (q1 )Φn2 (q2 ), where Φn (q) is the wave function of one-dimensional oscillator.
6
(12)
The interaction of an electromagnetic wave in the far infrared range with electrons confined in a QD can be described in a dipole approximation. For an electron, the interaction Hamiltonian is given by ~ 0 · ~r exp(iωt), Hinter = eE
(13)
where E0 is the amplitude of oscillations of the electric field. The Raman cross section can be given by the perturbation theory. The DCS for electron Raman scattering in a volume per unit solid angle for incoming light of frequency ωl , and scattered light of frequency ωs , is given by by Ref. [35] d2 σ V 2 ω 2 η(ωs ) = 3 s4 W (ωs , ~es ), dΩdωs 8π c η(ωl )
(14)
where η(ω) is the refraction index as a function of the radiation frequency, ~es is the unit polarization vector for the emitted secondary radiation, c is the light velocity in vacuum and W (ωs , ~es ) is the transition rate for the emission of secondary radiation (with frequency ωs and polarization ~es ), which is calculated according to Fermi’s golden rule: W (ωs , ~es ) =
2π X |Mf i |2 δ(Ef − Ei ). h ¯ f
(15)
In Eq. (14), Mf i and δ(Ef − Ei ) are defined by Mf i =
X hf |Hs |aiha|Hl |ii a
Ei − Ea + iΓa
,
(16)
and δ(Ef − Ei ) =
Γf . π[(Ef − Ei )2 + Γ2f ]
(17)
Here, |ii, |ai and |f i denote initial, intermediate and final states of the system with their corresponding energies Ei , Ea and Ef , respectively. These energies are 7
determinable by using the variational method. And Γf is the life-time width. The operator Hl is of the form
|e| Hl = m0
s
2π¯ h ~el · p~, V ωl
p~ = −i¯ h∇.
(18)
This operator describes the interaction with the incident radiation field in the dipole approximation. The Hamiltonian of the interaction with the secondary radiation field is given by |e| Hs = me (P, T )
s
2π¯ h ~es · p~, V ωs
p~ = −i¯ h∇.
(19)
Here ~el is the unit polarization vector for the incident radiation. This Hamiltonian describes the photon emission by the electron after transitions between conduction subbands of the system. The selection rules between the energy levels can be calculated from the dipole matrix elements and are as follows: polarization along the x or y axis: ∆n1 = ±1, ∆n2 = 0, and the y axis: ∆n1 = 0, ∆n2 = ±1. In this work, the DCS of Raman scattering for a three-level system is calculated for a QD using equation (13). We choose the backscattering configuration, where the incident radiation wave vector is parallel to the z axis, i.e., ki ||z, and the scattered radiation wave vector parallel to the z axis, i.e., kf ||z, with both the polarizations of the incident and scattered wave vectors parallel to the x axis and y axis, i.e., Z(X, Y )Z. In the initial state we have an electron in the Ψ01 state and an incident photon of energy h ¯ ωl with polarization along the x axis. From the Ψ01 state the electron out a transition to the intermediate state Ψ11 with energy Ea . From the intermediate state the electron undergoes a transition toward the Ψ10 state, emitting the laser
8
radiation as secondary radiation of energy h ¯ ωs with polarization along the y axis. 3 Results and discussions In this section we will calculate the influence of an external magnetic field, hydrostatic pressure, temperature and the dot parameters on the DCS of electron Raman scattering in a two-dimensional typical GaAs anisotropic QD. The lifetimes of the final and intermediate states are Γf = Γa = 1.0meV . All calculations of electron Raman scattering are performed in the scattering configuration Z(X, Y )Z. Fig. 1 shows the variation of the DCS as a function of the diffusion photon energy for three different values of the degree of anisotropy, i.e., Ωx /Ωy = 2.25, 4.0, and 9.0 (Ly /Lx = 1.5, 2.0, and 3.0), respectively. The QD size is defined as the confinement characteristic length R =
q
Lx Ly . In Fig. 1, the dot size R is set to be 6.0nm. Here,
hydrostatic pressure and magnetic field are set to be P = 20kbar and B = 10T , respectively. In Fig. 1, the DCS values of Ωx /Ωy = 4.0 and 9.0 should multiply 0.2 and 0.01 times, respectively. From this figure we can find that the degree effect of anisotropy of the QD on Raman scattering is significant. We observe that the rapid decrease in resonance peak of the DCS results from the increasing the anisotropy degree of QDs. And the resonance peak of the DCS moves towards lower energies (red shift) with increasing the ratio Ωx /Ωy . These results can be easily explained, when the ratio Ωx /Ωy increases, the confinement of the y direction will decrease and the confinement of the x direction will increase. This leads to that the energy difference Ef a between the Ψa and Ψf states decreases and the energy difference Eai between the Ψi and Ψa states increases. Hence, the resonant peaks of all DCS suffer an obvious red-shift and the peak intensities rapid decrease with increasing the ratio 9
Ωx /Ωy . It is possible to manage the resonance frequency and the amplitude of the DCS by controlling the anisotropy degree of QDs. In order to investigate the influence of geometrical size of QDs, we plot, in Fig. 2, the DCS as a function of the diffusion photon energy for three different dot sizes. In Fig. 2, the degree of anisotropy, hydrostatic pressure, and magnetic field are set to be Ωx /Ωy = 2.25, P = 20kbar, and B = 10T , respectively. The plots indicate that all peak positions of DCS shift to lower energies (red shift) with increasing R. This result is in good agreement with those obtained in spherical QDs [36, 37]. Figure 2 also shows that, as R is increased, the peak intensities of the DCS decrease. Similar behavior is also observed in Ref. [37]. Obviously, the confinements of the x and the y directions decrease with increasing dot size R so that the energy differences Ef a and Eia decrease. The resonant peaks of all DCS suffer an obvious red-shift and the peak intensities decrease with increasing R. Therefore, we can say that the smaller dot size should used in order to obtain the considerable changes in the Raman scattering of anisotropic QDs. Fig. 3 shows the effect of external magnetic field on the Raman scattering in a two-dimensional anisotropic QD. We plot the DCS as a function of the diffusion photon energy for three different strengths of magnetic field. In Fig. 3, the degree of anisotropy, hydrostatic pressure and dot size are set to be Ωx /Ωy = 2.25, P = 20kbar and R = 6.0nm, respectively. From this figure the magnetic field-induced effect on the Raman scattering of anisotropic QDs is clear. It is readily seen that the resonant peaks of all DCS suffer an obvious red-shift and the peak intensities decrease with increasing B. This result is in good agreement with that obtained in a cylindrical 10
QD [15]. The physical origin of these results is that the energy difference Ef a decreases and the energy difference Eai states increases with increasing B. This phenomenon tells us that by varying the strength of external magnetic field, it is possible to control the frequency shift and the peak intensity in the Raman spectrum of anisotropic QDs. It is well known that application of hydrostatic pressure results in a modification of the physical properties [38, 39]. This mainly is due to deformation of the interatomic bonds [40]. Furthermore, in order to investigate the influence of the hydrostatic pressure on the Raman scattering in anisotropic QDs, we plot, in Fig. 4, the DCS as a function of the diffusion photon energy for three different hydrostatic pressure values, i.e., P = 0, 20 and 40kbar, respectively. Here, the degree of anisotropy, dot size and magnetic field strength are set to be Ωx /Ωy = 2.25, R = 6.0nm, and B = 10T , respectively. As we can see from Fig. 4 that as a result of the increase in the pressure, the magnitude of the DCS decreases and the Raman spectrum shows a red shift. This is obtained due to the increase of the electron effective mass with increasing hydrostatic pressure. Therefore, an increase in the hydrostatic pressure leads to the more confinement of electrons in QDs and thus weakens the DCS of Raman scattering. The obtained results give a new degree of freedom in device applications, such as photo-detectors, electro-optical modulators, and all optical switches. In conclusion, we have investigated the effects of hydrostatic pressure, and external magnetic field on electron Raman scattering in a two-dimensional anisotropic QD. The study has considered the simultaneous effects of external magnetic field 11
and hydrostatic pressure for several values of the degree of anisotropy of QDs. With typical semiconductor GaAs based materials, the DCS as a function of the diffusion photon energy have been obtained. The results show that the electron Raman scattering in anisotropic QDs is strongly affected by the degree of anisotropy, dot size, and applied magnetic field strength and hydrostatic pressure of QDs. We found that the resonant peak of the DCS of Raman scattering in anisotropic QDs can be red or blue shifted by external probes such as applied magnetic field and hydrostatic pressure. The larger change of the Raman spectrum will be obtained by controlling the anisotropy degree of QDs. The dependence of DCS on size of the QD, the magnetic field and the hydrostatic pressure could be used for spectroscopic characterization of low-dimensional semiconductor systems. Acknowledgment: This work is supported by the National Natural Science Foundation of China under Grant No. 11074055.
12
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Figure captions: Figure 1 Dependences of the DCS of Raman scattering in a two-dimensional anisotropic QD as a function of photon energy for the three different Ωx /Ωy values with P = 20kbar, R = 6.0nm and B = 10T . Figure 2 Dependences of the DCS of Raman scattering in a two-dimensional anisotropic QD as a function of photon energy for the three different R values with P = 20kbar, Ωx /Ωy = 2.25 and B = 10T . Figure 3 Dependences of the DCS of Raman scattering in a two-dimensional anisotropic QD as a function of photon energy for the three different B values with P = 20kbar, Ωx /Ωy = 2.25 and R = 6.0nm. Figure 4 Dependences of the DCS of Raman scattering in a two-dimensional anisotropic QD as a function of photon energy for the three different P values with B = 10T , Ωx /Ωy = 2.25 and R = 6.0nm.
16
Ωx/Ωy=2.25
Figure 1
Ωx/Ωy=3.0
P=20.0kbar,B=10.0T,R=6.0nm
Ωx/Ωy=9.0
DCS (arb.units)
0.6
0.4
×0.2 0.2
×0.01
0.0
0
10
20
30
hωs (meV)
40
50
Figure 2
R=6.0nm R=7.0nm R=8.0nm
P=20.0kbar,B=10.0T,Ωx/Ωy=2.25
DCS (arb.units)
0.6
0.4
0.2
0.0
0
10
20
30
hωs (meV)
40
50
B=5.0T B=10.0T B=15.0T
Figure 3 1.0
P=20.0kbar,R=6.0nm,Ωx/Ωy=2.25
DCS (arb.units)
0.8
0.6
0.4
0.2
0.0
0
10
20
30
hωs (mev)
40
50
Figure 4
P=0kbar P=20.0kbar P=40.0kbar
B=10.0T,R=6.0nm,Ωx/Ωy=2.25
DCS (arb.units)
0.8
0.6
0.4
0.2
0.0
0
10
20
30
hωs (meV)
40
50
Highlights：
1. Electron Raman scattering process of a two-dimensional anisotropic quantum dot is investigated. 2. The larger peak intensity will be obtained by controlling the anisotropy degree. 3. It is possible to control the frequency shift by external probes such as magnetic field and hydrostatic pressure.
Ωx/Ωy=2.25 Ωx/Ωy=3.0 Ωx/Ωy=9.0
DCS (arb.units)
0.6
0.4
×0.2 0.2
×0.01
0.0
0
10
20
30
hωs (meV)
40
50