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Surface Review and Letters, Vol. 15, No. 3 (2008) 201–205 c World Scientific Publishing Company
EFFECTS OF MAGNETIC AND ELECTRIC FIELDS ON THE HYDROGENIC IMPURITY IN AN ELLIPSOIDAL PARABOLIC QUANTUM DOT E. KASAPOGLU∗,‡ , H. SARI∗ and I. SOKMEN† ∗ Cumhuriyet University, Department of Physics, 58140 Sivas, Turkey †
Dokuz Eyl¨ ul University, Department of Physics, 35160 Izmir, Turkey ‡
[email protected] Received 18 April 2007 The binding energy of a hydrogen-like impurity in an ellipsoidal parabolic quantum dot under the magnetic and electric fields have been discussed by using the effective mass approximation and the variational method. We have calculated the effects of the magnetic and electric fields on the binding energy of donor impurities in the quantum dots with different confinement potentials. We conclude that the structural confinement is very effective, and especially in the weak confinement potential case the magnetic field dependence of the donor binding energy is more pronounced. Keywords: Donor impurities; lens-shaped quantum dot. PACS Number(s): 73.21.La, 73.20.Hb
1. Introduction
over the years. Under different conditions, different geometric quantum dots can be confined; examples are rectangular boxes,10–13 spheres,14,15 disc or cylinders,16,17 spheroids,18,19 parabolic cylinders,20 and lens-shaped dots.21–26 The effects of applied magnetic and electric fields on the physical properties of low-dimensional systems are studied with the proposal of understanding the fascinating novel phenomena and of modeling new devices or improving the performance and tunability. There have been published numerous theoretical works on hydrogenic impurity states in the presence of the magnetic field.25,27–30 Quantum dots have been found to have a series of different ground states such as the magnetic field is increased.27 Therefore, theoretically understanding the nature of the electronic state of the hydrogenic impurities in the presence of the external fields is still one major purpose in these fields.
Recent research developments in crystal growth techniques, such as molecular beam epitaxy, metalorganic chemical-vapor deposition, and chemical lithography have made possible the fabrication of a wide variety of nanostructure semiconductor materials, where the electronic structure of the carriers plays an important role.1–4 Numerous studies have been devoted to various aspects of electronic states associated with low-dimensional semiconductor heterostructures such as quantum wells, quantum wires, and quantum dots. Quantum dots have attracted great interest due to their potential applications in a wide variety of optoelectronic and microelectronic devices. Quite different from that in the bulk materials, because of the reduced dimensionality, quantum confinement and novel transport behaviors can be observed.5–9 A wide variety of shapes have been made and theoretically studied 201
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E. Kasapoglu, H. Sari & I. Sokmen
In this work, we calculate the ground state energies of a hydrogenic donor impurity located at different positions in an ellipsoidal parabolic GaAs/In1−x Gax As quantum dot, under the action of the magnetic and electric fields using the effective-mass approximation within the variational approximation.
Lz , the terms associated with the motion in the x- and y-directions(Hx , Hy ), and the remainder of the term associated with Coulombic interaction between the donor electron and ionized impurity ion. So, the Hamiltonian can be written as follows: H = Hx + H y + H z + −
2. Theory We investigate the hydrogen-like impurity, which is confined in an ellipsoidal parabolic (GaAs/ In1−x Gax As) quantum dot under magnetic and electric fields. We suppose these fields directed along the z-axis; so the Hamiltonian will be written as e 2 m∗ 2 2 2 1 p − A(r ) + ω α (x + y 2 ) H= 2m∗ c 2 m∗ 2 2 ω z + eF z + 2 e2 − , (1) εo (x − xi )2 + (y − yi )2 + (z − zi )2 where εo is the dielectric constant, m∗ is the effective mass of the electron, ωis the frequency of the ellipsoidal parabolic quantum dot confinement potential r ) is in the (x–y) plane and along the z-axis, and A( the vector potential of the magnetic field and it can × r. r ) = 1B be expressed as A( 2 To consider the different shapes of the quantum dots, we define α as the symmetric parameter arising from difference of the confinement potential strength in the (x–y) plane and z-direction. We note that for α = 1, it corresponds to the spherical potential; otherwise, it corresponds to the ellipsoidal parabolic potential. In this case, the Hamiltonian reduces to ieB 1 (x∂/∂y − y∂/∂x) p2 − H= ∗ 2m c e2 B 2 2 m∗ 2 2 2 2 + ω α (x + y 2 ) (x + y ) + 4c2 2 +
m∗ 2 2 ω z + eF z 2
e2 , (2) − εo (x − xi )2 + (y − yi )2 + (z − zi )2 where the positions of the ionized donor atom and electron are given as (xi , yi , zi ) and (x, y, z), respectively. The Hamiltonian can be separated into different parts: the Hamiltonian associated with the motion in the z-direction (Hz ), the term involving
eB Lz 2m∗ c
e2 , εo (x − xi )2 + (y − yi )2 + (z − zi )2
(3)
where Hx,y =
p2x,y e2 B 2 2 m∗ 2 2 2 ω α xx,y + x + ∗ 2m 8m∗ c2 x,y 2
and Hz =
p2z m∗ 2 2 ω z + eF z. + 2m∗ 2
It can be seen that the operators Hx , Hy , Hz , and Lz commute with one another, and it is possible to choose wave functions which are simultaneously eigenfunctions of all operators. As an exact solution of the problem posed by Eqs. (1)–(2) is not available, we propose the following variational wave function: ψ(r ) = N ψ(x)ψ(y)ψ(z) √ 1 2 2 2 × e− λ (x−xo ) +(y−yo ) +(z−zo ) ,
(4)
where λ is the variational parameter, N is the normalization constant, ψ(x), ψ(y), ψ(z) are the exact solutions of Hx , Hy , and Hz , respectively. So, the binding energy can be obtained from the following: EB = Ex + Ey + Ez − ψ(r )| H |ψ(r ) ,
(5)
where Ex , Ey , and Ez are the ground state energies of the donor electron x-, y-, and z-directions in the absence of the impurity, respectively.
3. Results and Discussion The results are obtained using the following input parameters: m∗ = 0.067 m0 (m0 is the free electron mass), ε0 = 12.5 ( the static dielectric constant is assumed to be same everywhere). In Fig. 1 we present the variation of the binding energy of the hydrogenic impurities in a quantum dot for α = 1 and 0.25 values which correspond to the spherical and ellipsoidal parabolic quantum dot, respectively, as a function
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Effects of Magnetic and Electric Fields on the Hydrogenic Impurity in an Ellipsoidal Parabolic Quantum Dot
203
30 ω = 36meV
B = 20 T
BINDING ENERGY (meV)
25
α =1
B = 10 T
B = 20 T
α=1
1
20
0.75
B = 10 T
Ψ(x,y,0) 2 B=0
15
0.5
2
0.25 0 0
-2
α = 0.25
10
y
0
x
-2 2
5
(a) 0 0
50
100
150
200
IMPURITY POSITION ( )
α = 0.25
Fig. 1. For α = 1 and 0.25, ω = 36 meV, the variation of the impurity binding energy as a function of the impurity position under the magnetic field.
of the impurity position (0, 0, zi ) for several values of the magnetic field B with potential equal to 36 meV. This figure shows that the binding energy decreases as the impurity gets further from the center of the quantum dot. It is interesting to note that the effect of the magnetic field is more pronounced for the weak bound potential, α = 0.25, than for the stronger confinement potential, α = 1. We can explain this behavior as follows: it is well known that, in the range of the spatial confinement where the geometric confinement predominates, the effect of the magnetic field is not very pronounced. On the other hand, the electron wave function is softer and more squeezable in weak bound potential, which causes the magnetic-field-induced effect to be more dominant in those quantum dots. Also it should be noted that, the binding energy of the impurity with strong confinement is larger than that of the soft one; the differences come from the feature of probability distribution of the donor electron in the different bound potentials. To analyze this behavior more clearly, we give the square of the electron wave function in the (x–y) plane vs the coordinates x, y for α = 1, 0.25, in Fig. 2. One can clearly see from these figures that, in the strong confinement potential case, α = 1, the donor electron is mostly localized around the center of the dot and the probability of finding the donor
1 0.75
Ψ(x,y,0) 2
0.5
5
0.25 0 0 -5
y
0 -5
x
5
(b)
Fig. 2. At ω = 36 meV, the squared electron wave function in the (x–y) plane vs the coordinates x, y in the absence of the external fields for (a) α = 1 and (b) α = 0.25.
electron and impurity ion in the same plane is larger than that of the weak confinement potential case, α = 0.25, where the electron wave function is softer. Therefore, the binding energy in the spherical quantum dot, α = 1, is more pronounced than that of the ellipsoidal parabolic one, α = 0.25. To find the effects of different confinement on the impurity binding, the variation of the binding energy of the hydrogenic impurities in a quantum dot for α = 1, 0.25 as a function of the impurity position (0, 0, zi ) for B = 0, 10, and 20 T with ω = 16 meV is given in Fig. 3. By comparing the results obtained from Fig. 3 with that of Fig. 1, we see that as expected, in the strong confinement potential case (ω = 36 meV), the coulombic interaction between the electron and
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E. Kasapoglu, H. Sari & I. Sokmen 35
20
B = 20 T
ω = 36meV
ω = 16meV
B = 10 T
30
15
B = 20 T
BINDING ENERGY (meV)
BINDING ENERGY (meV)
B= 0
B = 10 T
B=0
10
25
20 B = 20 T 15
5
B = 10 T 10 B=0
0
5
0
50
100
150
200
0.0
0.2
0.4
IMPURITY POSITION ( )
impurity ion and the impurity position dependence of the binding energy is more pronounced and the binding energy is more sensitive to the magnetic field in the weak bondage potential case (ω = 16 meV), contrary to the case of the strong confinement. We show in Fig. 4 the binding energy as a function of the parameter α, which corresponds to the strength of the confinement, for B = 0, 10, and 20 T with ω = 36 meV. As expected, the binding energy increases with the increasing confinement parameter α. The effect of the parameter α is to squeeze the electron wave function along the x- and y-directions. This is a consequence of the increasing geometric confinement of the donor electron with α. Also we see that the magnetic field effect is more pronounced for small values of the parameter α which correspond to the soft confinement potential. It may be interesting to evaluate the electric field dependence of the impurity binding energy in the quantum dot with different confinement potentials. Figure 5 shows the variation of the binding energy of a donor located in the center of the quantum dot in the presence of the electric field as a function of the confinement parameter α. In the presence of the electric field, the coulombic interaction between the electron and the donor impurity located in the center of
0.6
0.8
1.0
Fig. 4. For ω = 36 meV, the variation of the binding energy of on-center impurity in the quantum dot with respect to the shape of the confinement potential under the magnetic field.
25 ω = 36meV
F=0 20 BINDING ENERGY (meV)
Fig. 3. For α = 1 and 0.25, ω = 16 meV, the variation of the impurity binding energy as a function of the impurity position under the magnetic field, where solid (dashed) curves are corresponding to α = 1 (α = 0.25).
α
F = 5 kV/cm 15
F = 10 kV/cm
10
5 0.0
0.2
0.4
α
0.6
0.8
1.0
Fig. 5. For ω = 36 meV, the variation of the binding energy of on-center impurity in the quantum dot with respect to the shape of the confinement potential under the electric field.
the quantum dot decreases. This behavior is related to the change of the electron confinement in the zdirection in the quantum dot. The electron shifts to the left side of the dot; thus the donor electron is
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Effects of Magnetic and Electric Fields on the Hydrogenic Impurity in an Ellipsoidal Parabolic Quantum Dot
mostly localized in the left side of the structure in the z-direction. Therefore, the probability of finding the electron and the donor impurity in the same plane gets to be smaller with the electric field resulting in a reduction in the binding energy. Also we see that for large values of the electric field, the confinement of potential dependence of the binding energy gets to be weak.
4. Summary As a result, we have calculated the effects of the magnetic and electric fields on the binding energy of donor impurities in the quantum dots with different confinement potentials with the use of a variational approximation. We conclude that the structural confinement is very effective and especially in the weak confinement potential case the magnetic field dependence of the donor binding energy is more pronounced. It turns out that the donor binding energy of spherical quantum dot is larger than that of the ellipsoidal parabolic quantum dot. We expect that the obtained results and the method used in this study will be helpful in the studies based on the effects of changing the geometry of the self-assembled semiconductor quantum dots.
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