1 s–2 p ± transitions between hydrogenic states in GaAs low-dimensional systems under external fields

Superlattices and Microstructures, Vol. 29, No. 6, 2001 doi:10.1006/spmi.2001.0985 Available online at http://www.idealibrary.com on

1s–2 p± transitions between hydrogenic states in GaAs low-dimensional systems under external fields E CATERINA C. N ICULESCU† ‘Politehnica’ University of Bucharest, Department of Physics, 313 Splaiul Independentei, RO-77206 Bucharest, Romania (Received 2 March 2001) A variational formalism for calculating the binding and the transition energies between 1s- and 2 p± -like states of a shallow donor in cylindrical pills of GaAs low-dimensional systems, under the action of the electric and magnetic fields applied in the axial direction, is presented. Results were obtained for several values of the magnetic field as a function of the system geometry and the applied electric field. Within the two-dimensional limits, the theoretical results for the 1s–2 p+ transition energies are in good agreement with infraredmagnetospectroscopy measurements on donor doped quantum-wells. c 2001 Academic Press

Key words: low-dimensional systems, binding energy, shallow donor, magnetic and electric fields.

1. Introduction The effects of applied external fields on the physical properties of low-dimensional systems constitute a subject of considerable interest from both the theoretical and technological point of view, due to the importance of these systems in the development of new semiconductor devices. Magnetospectroscopy experiments on shallow donor impurities in GaAs/GaAlAs multiple quantum-well structures were performed by Jarosik et al. [1]. Far-infrared measurements obtained by Yoo et al. [2] have enabled the observation of the effects of electric fields on the electronic states of impurities in selectively donor-doped GaAs/GaAlAs quantum-wells (QWs). The effects of magnetic and electric fields on the intradonor transition energies between the 1s-like ground state and 2 p+ -like excited state of hydrogenic donors in these structures were studied following a variational calculation within the effective-mass approximation [3]. Cen and Bajaj [4], on the other hand, have developed a method for the ‘dielectric quantum well’ in the presence of parallel electric and magnetic fields taking into account the interfacial dielectric constant mismatch. The Franz–Keldysh effect on Landau levels and magnetoexcitons in QWs with a strong dc electric field in the quantum-well plane and a crossed static magnetic field aligned with the growth direction has been investigated by Citrin and Hughes [5]. A number of studies of the binding energies, density of impurity states, transition energies, and photoluminescence spectra associated with shallow impurities in quantum-well wires (QWWs), and quantum dots (QDs) have been reported. Latge et al. [6] studied infrared transitions between hydrogenic states in cylindrical infinite GaAs/GaAlAs QWWs without applied external fields. Lomascolo et al. [7] investigated the competition between the exciton and free-carrier recombination mechanism by means of the magneto-optical † E-mail: [email protected]

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time resolved photoluminescence in lnx Ga1−x As/GaAs V-shaped quantum wires. Numerical calculations and magnetophotoluminescence studies of ln0.13 Ga0.8 As/GaAs modulated-barrier quantum wires have been performed in magnetic fields up to B = 9 T by Bayer et al. [8]. Recently, the effects of an applied magnetic field on the binding energy of donor impurities in cylindrical GaAs QWWs have been studied [9, 10]. It was shown that some excited states are not bounded for some values of the wire radius and of the magnetic field [9]. Also, as a result of the presence of the magnetic field, the impurity binding energy may be increased or decreased depending on the impurity location in the quantum structure [10]. The effect of the quantum confinement on the energy spectra has been studied in the case of neutral donors [11] and negatively charged donors [12] in spherical QD. Chiba and Ohnishi [13] have investigated quantum-confined Stark effects on the electronic structure of a GaAs microcluster and have analyzed the electric field-ionization mechanism for a confined exciton. The theoretical results by Montes et al. [14] and Duque et al. [15] show that the presence of the electric field in cylindrical GaAs low-dimensional systems (LDSs), together with the donor position, is determinant for the existence of bounded excited states in these structures. The optical properties of correlated electrons in quantum dots with small confinement in a strong magnetic field [16] and the effect of free carriers on photoluminescence from modulation-doped self-assembled QDs in a magnetic field [17] have been studied by Wojs et al. In this work we study the effects of electric and magnetic fields, both applied parallel to axis of a cylindrical GaAs LDS, on the binding and transition energies between the 1s-like ground state and 2 p± -like excited states of a donor impurity.

2. Theory 2.1. Impurity located at the center In the effective-mass approximation, the Hamiltonian of a shallow donor located at the center of a cylindrical LDS under the action of electric and magnetic fields applied in the axial direction (chosen as the z direction) is H = H0 + Hc .

(1)

  e ¯ 2 1 H0 = p¯ − A + V (ρ, z) + |e|F z 2m ∗ c

(2)

Here

is the subband Hamiltonian and e2 (3) εr is the Coulomb interaction term. We have used the standard notation for each term [3]. The potential vector AE is selected as AE = 12 ( BE × rE) and the well potential is defined as  0, ρ < R, |z| < L/2 V (ρ, z) = , (4) ∞, ρ ≥ R, |z| ≥ L/2 Hc = −

where R and L are the LDS radius and length, respectively. 2 If the effective Bohr radius a ∗ = mε∗h¯e2 is the length unit and the effective Rydberg Ry ∗ = energy unit, the Hamiltonian becomes H = −∇ 2 − iγ Here F˜ =

eFa ∗ R∗

γ 2ρ2 2 ∂ + + V (ρ, z) + F˜ z − . ∂ϕ 4 r

is the strength of the applied electric field and γ =

eh¯ B 2m ∗ c R ∗

m ∗ e4 2ε2 h¯ 2

is the

(5) is the energy of the first

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Landau level. For a hydrogenic donor in GaAs, a ∗ = 100 Å, Ry ∗ = 5.72 meV, F˜ = 1 corresponds to F = 5.72 kV cm−1 , and γ = 1 to B = 6.76 T. We treat the F˜ z and Hc terms in the Hamiltonian variationally and proceed to determine the energy levels in a two-step procedure. In the absence of the quantum confinement the calculation of the eigenstates for a particle subject to a magnetic field is, in principle, an exact solvable problem [18]. Using the method of separable variables, the ground electronic wave function of the Hamiltonian in the absence of the impurity and without electric field is expressed as   πz 80 (ρ, z) = S(ρ) cos , (6) L where S(ρ) is the exact solution of the cylindrical QWW with magnetic field [10]  exp(−ξ/2)U (u, 1, ξ ), ρ < R S(ρ) = 0, ρ ≥ R. E0

(7)

E1 1 Here ξ = γρ 2 , u = ( 2γ0 − 2γ − 2 ) with E 1 = πL 2 , and U (u, 1, ξ ) is the confluent hypergeometric function. E 00 is the ground state energy of the LDS with the magnetic field. It is obtained by using the boundary condition 2

80 (R, z) = 0.

(8)

We assume that the subband wave function is modified by the electric field as follows [19]: 8(ρ, z) = exp(−ηz)80 (ρ, z)

(9)

and the lowest subband energy E 0 is determined from minimization of the expression hE 0 (η)i =

h8|H0 |8i h8|8i

(10)

with respect to η. The variational 1s- and 2 p± -like donor wave functions are taken as products of the solution 8(ρ, z) of the LDS in the presence of the electric and magnetic fields and hydrogenic-like functions 1s and 2 p± [3] are given as:  q  0nlm (E r ) = ρ |m| exp(imϕ) exp − ρ 2 + z 2 /λ , (11) where λ is a variational parameter. The binding energies of the donor states are obtained as E b = E 0 − min λ

h8(ρ, z)0nlm (E r )|H |8(ρ, z)0nlm (E r )i . h8(ρ, z)0nlm (E r )|8(ρ, z)0nlm (E r )i

(12)

In Figs 1 and 2 we present the binding energy for the ground and 2 p− -like states in a GaAs LDS with R = 2a ∗ as a function of the axial length and for different values of the applied fields. We varied the length of the structure in order to simulate QD-like and QWW-like structures. It is apparent that as the dimensions of the structure are reduced (QD-limit), the binding energy of the studied states is enhanced due to an increase in the electrostatic interaction between the electron and the impurity. As expected, in the presence of the electric field the binding energy of the on-center donor decreases, due to the displacement of the maximum of the probability density to negative values in the axial direction. The electric-field-induced polarization effect is stronger for the ground state because the 1s-like wave function has nonzero values at the origin. Since the magnetic field provides an additional confinement in the transverse directions, a large increase in the energies with B at high magnetic fields is observed. In comparing Figs 1 and 2, we observe that the magnetic field affects the 2 p− state more strongly than 1s state, due to the wider spatial extension of the 2 p− wave functions. As Figs 1 and 2 show, the effects of the externally applied fields are more pronounced for large L (QWW-limit), when the geometric confinement is a small perturbation on the electric and magnetic

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Superlattices and Microstructures, Vol. 29, No. 6, 2001 1s-like R = 2a*

3

A

B=0 T

Eb (Ry*)

0 kV/cm

2 50 kV/cm

1

10 kV/cm

100 kV/cm

0

2

4

6 L(a*)

1s-like R = 2a*

3

8

B

10

B = 15 T

Eb (Ry*)

0 kV/cm

2 50 kV/cm

1

10 kV/cm 100 kV/cm

0

2

4

6 L(a*)

8

10

Fig. 1. Binding energy of the 1s-like state of a donor impurity at the center of a cylindrical GaAs LDS with R = 2a ∗ as a function of the axial length and for different values of applied electric field. A, in zero magnetic field; B, in a 15 T magnetic field.

terms. Note that R = 2a ∗ in the zero-field limit (B = F = 0). Our results (Fig. 1A) agree very well with those obtained by Duque et al. [15]. Notice that, in the absence of the magnetic field and for small values of the radius (R < 3a ∗ ), the 2 p− like state of an on-center donor in a cylindrical GaAs LDS is unbounded, as has already been pointed out by Villamil et al. [9]. The appearance of the 2 p− -like bounded state is allowed at high magnetic fields for a wider range of axial lengths, as is shown in Fig. 2. Fig. 2 also exemplifies the competition between the effects of the electric field and the quantum and magnetic confinement. Beyond a critical axial length value (which is F-dependent), electric-induced polarization predominates and the 2 p− -like state of the impurity becomes unbound. For a given F, the value of the critical length diminishes with the decreasing of B, due to a weaker magnetic confinement in the radial direction. The effects of the applied electric field on the intradonor 1s–2 p± transition energies E t = E b1s − E b2 p± are shown in Fig. 3 for a GaAs LDS with R = 2a ∗ and different values for the axial length. It is observed that over the studied range of F the energy differences are only weakly dependent upon the electric field, particularly in the QD limit (L = 2a ∗ ). However, the transition energies show a substantial variation with the applied magnetic field. We found that 1s–2 p-transition energy decreases and 1s–2 p+ transition energy increases as functions of the magnetic field, in agreement with the results obtained by Villamil et al. [9]. In the absence of the calculations of donor binding energies or related experimental data on transition energies involving donor states in these particular structures, the results from these calculations have been compared with corresponding calculations for QW structures. In order to simulate a 2D system, we chose a cylindrical LDS with R  L. In Fig. 4 we present the binding energy for the 1s- and 2 p− -like states of

Superlattices and Microstructures, Vol. 29, No. 6, 2001

Eb (Ry*)

0.4

389

2p−-like R = 2a*

B=5 T

0 kV/cm

0.2

50 kV/cm

0.0

A

10 kV/cm

100 kV/cm

2

1.5

4 L (a*)

6

2p−-like R = 2a*

B

B = 15 T

0 kV/cm

Eb (Ry*)

1.0 10 kV/cm 100 kV/cm

0.5

0.0

2

4

50 kV/cm

6 L (a*)

8

10

Fig. 2. Binding energy of the 2 p− -like state of a donor impurity at the center of a cylindrical GaAs LDS with R = 2a ∗ as a function of the axial length and for different values of applied electric field. A, in a 5 T magnetic field; B, in a 15 T magnetic field.

an on-center donor in a GaAs cylindrical LDS with R = 30a ∗ and L = 5a ∗ as a function of the electric field and for different values of the applied magnetic field. Note that: (i) in the 2D limit, the 2 p− -like state is bound independently of the intensity of the magnetic field and in the whole range of the electric field that we have considered in our calculations (F ≤ 25 kV cm−1 ); (ii) the spread of the binding energies as the applied fields vary is larger in 2D limit than in 0D and 1D structures, because of the weaker quantum confinement. Variation of the 1s–2 p+ transition energy as a function of the electric field for GaAs LDS, such as described above, is shown in Fig. 5. Experimental far-infrared magnetospectroscopy data reported by Yoo et al. [2] for an L = 5a ∗ GaAs QW are also presented. The agreement between the calculated and experimental results indicates that the values obtained in Ref. [2] are associated with 1s–2 p+ transition energy for donors close to the QW on-center position. Consequently, our results are compared with the theoretical values calculated by Latge et al. [3] for an on-center donor in an L = 5a ∗ GaAs QW. Comparison shows that, in the range of electric field studied, our method leads to better results. Even though this work has been performed for GaAs LDSs using the infinite potential model, for a weak quantum confinement regime (R  1a ∗ , L > 1a ∗ ), the approximation is justified. It seems probable that for high values of magnetic field strength (γ > 1), the donor envelope-function chosen by Latge et al. [3] as products of the exact solution of the square with electric field and hydrogenic-like function 0nlm is too simplistic. 2.2. Off-center impurity In the cases investigated above we considered a centrally positioned donor situation which can hardly be realized experimentally. The problem of an off-center donor is more general and leads to a series of

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Transition energy (Ry*)

4

R = 2a* 1s-2p−

A

3

B=0 T

2 B = 15 T

1

0

0

25

50 F (kV/cm)

75

100

R = 2a* B=5 T

B

Transition energy (Ry*)

4 1s-2p+

3

1s-2p−

2

1

0

25

Transition energy (Ry*)

6.5

50 F (kV/cm)

75

R = 2a* B = 15 T

100

C

6.0

1s-2p+

5.5

5.0

0

25

50 F (kV/cm)

75

100

Fig. 3. Intradonor 1s–2 p± transition energies for cylindrical GaAs LDSs with R = 2a ∗ and different axial lengths: L = 2a ∗ (solid lines); L = 4a ∗ (dashed lines); L = 10a ∗ (dot-dashed lines). Results are given as functions of the electric field at various magnetic field values.

effects that do not exist in nanostructures with an impurity localized in the center [2, 4, 11, 15]. Particularly in a strong confinement regime, a large spread of the ground-state level is expected as the donor position varies [11, 14]. It is also well known that the presence of a magnetic [10] or an electric field [14, 15] breaks down the degeneracy of the states corresponding to symmetrically positioned impurities in the structure. Thus, to calculate the states of an impurity located along the z-axis, a distance z i from the center of the LDS, the Coulomb interaction is rewritten as e2 Hc = − p ε ρ 2 + (z − z i )2

(13)

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391

Eb (Ry*)

2.0 1.5 9T

9T 7T

1.0

7T 0T

0.5 0.0

0T

0

5

10 15 F (kV/cm)

20

25

Fig. 4. Electric-field dependence of the 1s binding energy (solid lines) and 2 p- binding energy (dashed lines) for on-center impurity in a cylindrical GaAs LDS with R = 30a ∗ and L = 5a ∗ at various magnetic fields.

Transition energy (Ry*)

3.8

1s-2p+

3.4 9T

3.0

2.6 7T

2.2

0

5

10 15 F (kV/cm)

20

25

Fig. 5. On-center intradonor 1s–2 p+ transition energy versus applied electric field and for two magnetic field strengths. Solid lines: present work, for a cylindrical GaAs LDS with R = 30a ∗ and L = 5a ∗ . Dashed lines: theoretical results for an L = 5a ∗ GaAs QW (Ref. [3]). Full dots: experimental results for an L = 5a ∗ GaAs QW (Ref. [2]).

and hydrogenic-like functions 1s and 2 p± are given by:   q 0nlm (E r ) = ρ |m| exp(imϕ) exp − ρ 2 + (z − z i )2 /λ .

(14)

Figure 6 shows the binding energy of the ground state and the intradonor 1s–2 p+ transition energy as functions of the impurity position along the axial direction of a cylindrical GaAs LDS with R = 20a ∗ and L = 5a ∗ , for different values of the externally applied fields. Contrary to the results reported by Latge et al. [3] for QWs, where they found that for high electric fields the curve of E 1 shows a minimum around z i = 0.1L, we found that in all studied cases the transition energy monotonically decreases with the donor position as z i > 0. This is due to the fact that the 1s-like state, having the highest probability density around the center, changes more remarkably along with changing z i than the 2 p+ -like state. Therefore, the dependence of the 1s–2 p+ transition energy on the donor position is essentially determined by variation of the 1s binding energy with donor position, as Fig. 6 shows. From the same figure it is also observed that for each value of the electric field the ground state binding energy presents a maximum. As the electric field is increased, the maximum shifts to the interface z i = −L/2, since the charge distribution is displaced

392

Superlattices and Microstructures, Vol. 29, No. 6, 2001 20 kV/cm

2.5

1s B = 7T

10 kV/cm 0 kV/cm

2 Eb (Ry*)

3.5

A Transition energy (Ry*)

3

1.5 1 0.5 − 0.25

0 zi/L

3

C

10 kV/cm 0 kV/cm

Eb (Ry*)

2 1.5 1 0.5 0 − 0.5

− 0.25

0 zi/L

0.25

0.5

B

0 kV/cm

2.5

2 − 0.5

0.5

1s-2p+ B = 7T

10 kV/cm

3

4

1s B = 9T

20 kV/cm

2.5

0.25

Transition energy (Ry*)

0 − 0.5

20 kV/cm

− 0.25

0 zi/L

20 kV/cm 10 kV/cm

0.25

1s-2p+ B = 9T

0.5

D

0 kV/cm

3.5

3

2.5 − 0.5

− 0.25

0 zi/L

0.25

0.5

Fig. 6. 1s binding energy and 1s–2 p+ transition energy for a cylindrical GaAs LDS with R = 30a ∗ and L = 5a ∗ versus impurity position along the z-axis for different values of the applied axial fields.

toward the left. This effect is well known for QWWs without a magnetic field [14, 15]. We observe that, for impurities placed at z i < 0 1s–2 p+ transition energy varies in a similar way, in agreement with the results reported by Latge et al. [3]. As the magnetic field increases (Fig. 6C and D), implying greater electron confinement around the z-axis, the binding and transition energies increase but dependence on the donor position is similar. In conclusion, an approach for calculating the effects of both electric and magnetic fields on the binding and transition energies between the 1s-like ground state and 2 p± -like excited states of a hydrogenic donor impurity in a cylindrical LDS has been presented. We used effective-mass approximation within a two-step variational scheme. We found that the external fields applied can greatly change the electronic levels of the cylindrical GaAs LDS with proper R and L, and then the infrared-absorption spectra. The concept might be useful for designing some devices in the future. Also, in the two-dimensional limit, the calculated 1s–2 p+ transition energies compare well with experimental results on donor-doped quantum wells.

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