Appl. Phys. A 78, 101–105 (2004)
Applied Physics A
DOI: 10.1007/s00339-002-1955-x
Materials Science & Processing
e. kasapoglu1,u h. sari1 2 ¨ i. sokmen
Effect of crossed electric and magnetic fields on donor impurity binding energy 1 Cumhuriyet 2 Dokuz
University, Physics Department, 58140 Sivas, Turkey Eylül University, Physics Department, Izmir, Turkey
Received: 5 June 2002/Accepted: 21 August 2002 Published online: 17 December 2002 • © Springer-Verlag 2002 ABSTRACT By using an appropriate coordinate transformation, we have calculated variationally the ground state binding energy of a hydrogenic donor impurity in a quantum well in the presence of crossed electric and magnetic fields which are applied tilted at an angle to the layers. The dependence of the donor impurity binding energy on the well width, on the strength of the electric and magnetic fields, on the impurity position and on the directions of the external fields is discussed. PACS 71.55.Eq;
in the presence of crossed electric and magnetic fields. The electric and magnetic fields are directed perpendicular to each other but both are tilted with respect to the layers. We are unaware of any previous work to study the combined effects of an electric and a magnetic field on impurity states where both fields are tilted with respect to the layer. In our calculations, we obtained the explicit dependence of the impurity binding energy on the well width, on the position of the impurity, on the strengths of the electric and magnetic fields, and on the tilt angle.
71.55.-i 2
1
Introduction
In recent years many theoretical and experimental investigations have been performed on the issue of the hydrogenic binding of an electron to a donor impurity confined within a low-dimensional heterostructure [1–14]. An understanding of the electronic and optical properties of impurities in such systems is important because the optical and transport properties of devices made from these materials are strongly affected by the presence of shallow impurities. Also, magnetic and electric fields are effective tools for studying the properties of impurities in heterostructures. So, the use of the combined effects of an electric and a magnetic field on impurity states is also of interest, but little has been studied on this subject. Cen and Bajaj [4, 15, 16] have used a variational method to consider the impurity states in symmetric and asymmetric quantum wells subject to parallel electric and magnetic fields directed perpendicular to the heteroplanes. Wang and Chuu [17, 18] have calculated free-electron energy levels in single and double asymmetric quantum wells in the presence of a crossed electric field perpendicular to the heteroplanes and a magnetic field in the plane. Monozon et al. [19] have shown that an analytical approach can be used for the consideration of an impurity electron (or hole) in an infinite quantum well in the presence of crossed electric and magnetic fields. In this study, we report a calculation, performed with the use of a variational approximation, of the ground state binding energy of a hydrogenic donor impurity in a quantum well u Fax: +90-346/219-1186, E-mail:
[email protected]
Theory
We define the z axis to be along the growth axis, and take the magnetic field to be applied in the x –z plane at an angle θ to the x axis, i.e. B = (B cos θ, 0, B sin θ), and the electric field to be applied in the x –z plane at an angle θ to the z axis, i.e. F = (−F sin θ, 0, F cos θ). We choose a gauge for the magnetic field in which the vector potential A is written in the form A = (0, x B sin θ − z B cos θ) using the ∇.A = 0 gauge. Within the framework of an effective-mass approximation, the Hamiltonian of a hydrogenic donor in a GaAs quantum well, in the presence of an applied electric and magnetic field, can be written as H=
2 1 e e2 p + A(re ) + e Fre − + V(z e )x , (1) 2m e c ε0 |re − ri |
where m e is the effective mass, e is the elementary charge, p is the momentum, ε0 is the dielectric constant, and V(z e ) is the confinement potential profile for the electron in the z -direction. The functional form of the confinement potential is given by V(z e ) = V0 [S(z L − z e ) + S(z e − z R )] ,
(2)
where S is the step function, and the left and right boundaries of the well are located at z = z L = − L20 and z = z R = L20 , respectively. By using the following transformation, z cos θ = x sinθ
− sinθ cosθ
z x
(3)
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In the above equation α is a variational parameter; we have also used this function in our previous studies [20, 21]. Here 1 2 e B 2 1 2 px + p2z + z + e Fz H= py + (4) ψ(x ) and ψ(z ) are the wave functions of the electron in the 2m e 2m e 2m e c x and z directions, respectively, which are obtained exactly from the Schrödinger equation in these directions. 2 e The total energy of the system is evaluated by minimizing + V x e , z e − , 2 2 the expectation value of the Hamiltonian in (9) with respect ε0 x e − x i + (y − yi )2 + z e − z i to α: eB where the term m e c z p y is equal to zero z , p y = 0 . min ψ H˜ ψ = E˜ . (12) α After the coordinate transformation, the left and right boundaries of the wells on the x and z axes are The binding energy of the donor impurity ground state is given by x L,R = z L,R sin θ + x cos θ , (5) E˜ B = E˜ 0 − E˜ , (13) z L,R = z L,R cos θ − x sin θ , the Hamiltonian can be written as below: 2
2
respectively. The solution of the corresponding Schrödinger equation is not straightforward, since, after the coordinate transformation, the potential energy of the electron in the well V(x e , z e ) couples the variables x and z . In order to decompose the potential energy of the electron, we rewrite the step functions in (2) as follows: S(z L − z e ) = cos
2
θ S(z L − z e ) + sin2
θ S(x L − x e )
,
S(z e − z R ) = cos2 θ S(z e − z R ) + sin2 θ S(x e − x R ) .
(6)
By considering the above equations, we can separate the potential as V(x e , z e ) = V(x e ) + V(z e ) ,
(7)
where
V(x e ) = V0 sin2 θ S(x L − x e ) + S(x e − x R ) ,
V(z e ) = V0 cos2 θ S(z L − z e ) + S(z e − z R ) .
(8)
where E˜ 0 is the lowest electron total subband energy in the x and z directions. Substituting the expectation value of the Hamiltonian into (12), we obtain the ground state binding energy of the donor impurity. 3
Results and discussion
The values of the physical parameters used in our calculations were m e = 0.0665 m0 (where m 0 is the free-electron mass), ε0 = 12.58 (the static dielectric constant is assumed to be the same for GaAs and GaAlAs) and V0 = 228 meV. These parameters are suitable for GaAs/ Ga1−x Alx As heterostructures with an Al concentration of x∼ = 0.3. Without losing generality, and for simplicity in numerical calculations, we have chosen the boundaries of the well at x L(R) = ±( L20 ) sin θ and z L(R) = ±( L20 ) cos θ , after a coordinate transformation which satisfies the following equation derived from (5): z R − z L = L 0 cos θ ,
x R − x L = L 0 sin θ .
(14)
The binding energy of a donor impurity located (1) in the centre, (2) at the left side and (3) at the right side of a GaAs quantum well subjected to crossed electric and magnetic fields, versus the well width, for a tilt angle θ = 30◦ and two different electric-field values, is shown in Fig. 1a for B = 1 T and in Fig. 1b for B = 10 T. As seen in Fig. 1a, the 2 2 d d H˜ = − 2 + V˜ x˜e − 2 + V˜ z˜e + F˜ z˜ (9) binding energy of a donor impurity at the well centre increases dx˜ dz˜ as the well width L 0 increases, since the probability of find2 2 2 ing the electron around the impurity increases and reaches e B h 2 d2 2 + z − − , ˜ a maximum value. After a certain L 0 value ( L 0 ∼ = 100 Å), 2 2 4m 2e c2 R2y d y˜2 x˜e − x˜i + y˜ 2 + z˜e − z˜i the impurity binding energy begins to decrease with increasing L 0 . As L 0 increases, the localization of the electron in the well increases, but the probability of finding the elecwhere F˜ = FaB /10R y is the dimensionless electric field. We propose the following variational trial wave function tron and impurity in the same plane decreases. In the range for the electron bound to the impurity: 50 Å ≤ L 0 ≤ 100 Å, the impurity binding energy is independent of the electric-field value. As is known, the electron ψ = ψ(x˜ )ψ(z˜ )ϕ(y,α) . (10) becomes more energetic in narrow wells, and therefore the impurity binding energy is not affected by the electric field in Here the wave function in the y-direction ϕ(y, α) is chosen to this situation. As L increases ( L ≥ 100 Å), an electron that 0 0 be a Gaussian-type orbital function: approaches the well bottom begins to become sensitive to the 1/4 electric field strength. The Coulombic interaction between the 1 2 −y2 /α2 ϕ(y,α) = √ e . (11) electron and a donor impurity located in the centre decreases α π since the electron shifts to the left side of the well as the all lengths to the effective Bohr radius By scaling 2 2 a = ε h /m B 0 e e and the energies to the effective Rydberg 4 2 2 R y = m e e /2ε0 h and considering the above results, we can rewrite the dimensionless Hamiltonian of the system as
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Effect of crossed electric and magnetic fields on donor impurity binding energy
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that the impurity binding energy does not change except at large L 0 values. In Fig. 2, for θ = 45◦ , we show the impurity binding energy as a function of the well width for different impurity positions and electric-field values, and for magnetic-field values B = 1 T and 10 T. For this tilt angle value, the electron is subject to the effect of the same geometric confinement in both directions since the effective well widths and potential heights for the electron in both the x and the z directions are equal, and the donor impurity binding energy becomes maximum. For example, for L 0 = 50 Å, the change of the binding energy of a donor impurity located in the centre between θ = 30◦ and θ = 45◦ is approximately 1R y (∼ 5 meV). In cases 2 and
Binding energy of a donor impurity located (1) in the centre, (2), at the left side and (3) at the right side of a GaAs quantum well subject to crossed electric and magnetic fields, versus well width, for a tilt angle θ = 30◦ , two different electric-field values and two different magnetic-field values: a B = 1 T; and b B = 10 T FIGURE 1
electric field increases in magnitude. So, for F = 50 kV/cm, the impurity binding energy becomes weaker than that at F = 10 kV/cm, in contrast to the case of a donor impurity located at the left side of the well (curve 2 in Fig. 1). Naturally, a lower binding energy is obtained for an impurity located at the right side of the well (curve 3). In Fig. 1b, we show the effect of the magnetic field on the donor impurity binding energy in a quantum well with the same physical parameters as in Fig. 1a but with the magnetic field changed. For B = 10 T, when we compare the result with the previous case, we see
For θ = 45◦ , the variation of the impurity binding energy as a function of the well width for different impurity positions, electric-field values and magnetic-field values: a B = 1 T; and b B = 10 T FIGURE 2
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3, this change is approximately 0.6R y (∼ 3.5 meV). In addition, at this tilt angle, the impurity binding energy in the range 50 Å ≤ L 0 ≤ 100 Å begins to become sensitive to the electric field but is largely independent of the magnetic field. For θ = 60◦ , the dependence of the impurity binding energy on the well width, on the impurity position within the quantum well and on the magnitude of the external field is shown in Fig. 3a and b for B = 1 T and B = 10 T, respectively. At this value of the tilt angle, the probability of finding the electron in the barrier increases in the z direction, since the effective well width in this direction is very narrow. As seen in this figure, the impurity binding energy for all L 0 values begins to become sensitive to the electric field, since
the confinement by the external field predominates when the geometric confinement becomes weak. The binding energy of a donor impurity located in the centre of the quantum well as a function of tilt angle is shown in Fig. 4a and b for L 0 = 50 Å and 200 Å, respectively, for different values of the magnetic and electric fields. As seen in Fig. 4a, in the range 15◦ ≤ θ ≤ 45◦ , the donor binding energy increases as the tilt angle increases, independently of the external fields, and reaches a maximum value at θ = 45◦ . In this range, the change in the binding energy of a donor impurity located in the centre of the quantum well between θ = 15◦ and θ = 45◦ is approximately 3R y (16 meV), the Coulombic interaction between the electron and the donor impurity de-
For θ = 60◦ , the dependence of the impurity binding energy on the well width, on the impurity position within the quantum well and on the magnitudes of the external fields: a B = 1 T; and b B = 10 T
FIGURE 4 Binding energy of a donor impurity located in the centre of the quantum well versus tilt angle for different magnetic- and electric-field values: a L 0 = 50 Å; and b L 0 = 200 Å
FIGURE 3
KASAPOGLU et al.
Effect of crossed electric and magnetic fields on donor impurity binding energy
pends entirely on the geometric confinement, and the impurity binding energy decreases for angles larger than θ = 45◦ . For θ ≥ 45◦ , since the geometric confinement becomes weak, the impurity binding energy begins to become sensitive to the external fields (especially the electric field). For large L 0 values (Fig. 4b), the impurity binding energy depends on the external fields at nearly all tilt angles, since the geometric confinement becomes weak at large well widths. In the range 15◦ ≤ θ ≤ 45◦ , the change of the binding energy of the donor impurity between θ = 15◦ and θ = 45◦ is approximately 1R y (6 meV). 4
Conclusions
In this study, we have calculated, using a variational approximation, the ground state binding energy of a hydrogenic donor impurity in a quantum well in the presence of crossed electric and magnetic fields which are tilted with respect to the layers. In our calculations, we obtained the explicit dependences of the impurity binding energy on the well width, on the position of the impurity, on the strengths of the electric and magnetic fields and on the tilt angle. As seen in our results, the tilt angle causes important changes in the impurity binding energy. For example, we can say that the quantum well structure is reduced to a quantum wire in the case of the results obtained at θ = 45◦ , since the system is subject to the effect of the same geometric confinement in both the x and the z directions (i.e. the effective well
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widths and potential heights for the electron in both the x and z directions are equal ). We hope that these results will provide important improvements in device applications. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
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