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Localized donor states in axially symmetrical heterostructures
Superlattices and Microstructures 48 (2010) 288–297
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Localized donor states in axially symmetrical heterostructures W. Gutiérrez ∗ , L.F. García, I.D. Mikhailov Escuela de Física, Universidad Industrial de Santander, A.A. 678, Bucaramanga, Colombia
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Article history: Received 19 February 2010 Received in revised form 13 May 2010 Accepted 17 June 2010 Available online 18 July 2010 Keywords: Semiconductors nanostructures Neutral donor Binding energy Excited states Nanowire superlattice
abstract We present a general method for calculating the energy spectrum of donors confined in heterostructures with axial symmetry in the presence of magnetic and electric fields applied along the symmetry axis. The donor’s wave functions are chosen as a product of the Slater orbitals and an envelope function that is a solution of a one-dimensional differential equation, which we derive starting from Schrödinger’s variational principle. We calculate the energies of the ground and some excited states of a donor confined in multiple quantum wells and a nanowire superlattice as functions of the donor position, electric and magnetic fields for structures with different numbers and widths of the wells and the barriers. Our method could be applicable to a variety of complex quantum-confined semiconductor structures for which more rigorous approaches require extensive numerical calculations. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction For the past few decades different quasi-two-dimensional semiconductor structures, such as single and double quantum wells (DQWs), multiple quantum wells (MQWs), superlattices (SLs) and nanowire superlattices (NWSLs), have been studied extensively both theoretically and experimentally, and applied to various electronic and photonic devices. Taking electric and magnetic fields applied along to the growth direction together with the effects of the confinement and tunneling across the potential barriers, we have an interesting physical system in which the competition of these factors allows us to manipulate its electro-optical properties conveniently for applications in spintronics, optoelectronics, photovoltaics, and quantum information technologies [1].
∗
Corresponding author. Tel.: +57 7 6323095; fax: +57 7 6332477. E-mail address: [email protected] (W. Gutiérrez).
0749-6036/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2010.06.016
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The widespread interest in semiconductor structures with axial symmetry is related to the ability to manipulate the energy splitting between the lowest-lying states or the spatial distribution of carriers by varying the radius (in the case of the NWSL), thicknesses and number of wells and barriers. The electrical and optical properties of these structures also may be modified by impurities due to alteration of the carriers’ spatial distribution, to shift and to splitting of the low-lying energy levels in their presence. The resulting alteration of the electro-optical properties of the heterostructures depends on various factors, such as the donor position, the quantum confinement, the interwell tunneling and the electron–ion attraction. For this reason the elaboration of a simple and accurate method of calculation of the energy spectrum of donors (D0 ) confined in different types of the heterostructures has been the subject of intense research since the advent of nanotechnology. The variational [2], Monte Carlo [3], finite difference [4] and other methods [5] have been used for calculating the low lying energy levels of D0 confined in various types of heterostructures, such as QW [6], MQW [7], QWW and SL [8]. Nevertheless, the elaboration of a unified approach to the theoretical analysis of the localized donor states common to all heterostructures has been difficult due to the additional constraints that are imposed on the electron wave function at short distances by the singularity of the Coulomb potential. If the electron approaches the ion then the potential becomes large and negative. This must be canceled by a corresponding positive divergence in the kinetic energy. This condition can be satisfied only if the corresponding wave functions have a ‘‘cusp’’ at the point of the donor location, identical for all heterostructures independently of their geometries. Any accurate algorithm should take into account this circumstance in order to ensure a rapid convergence of the estimated D0 energies to their exact values. In this work a particular emphasis is given to the choice of the optimal wave functions for localized donor states which satisfy the ‘‘cusp’’ condition. We were motivated to analyze this subject by the interest in the elaboration a universal technique for calculating the energies and wave functions of the localized states in nanostructures induced by the presence in them of a shallow donor. Our main result is a general one-dimensional differential equation for the envelope function derived in Section 2, which allows us to analyze, in a simple form, modifications of the hydrogenic states due to the confinement in any axially symmetrical heterostructure. In Section 3 we compare results obtained with our method for ground state donor energies in different types of heterostructures, such as QW, MQW and SL, with corresponding results obtained previously by using other, more sophisticated techniques. We analyze dependencies of the donor binding energy in these structures on the geometric parameters, donor position, and intensities of the electric and the magnetic fields. Also, we present novel results for some donor s-like states in QW and NWSL. 2. Theory The object of our analysis is the dimensionless Hamiltonian H = H (0) (Q ) + V (z ) + α z ;
γ˜ = γ + 2
2
H (0) (Q ) = −∆ − p
2Q
ρ 2 + (z − ξ )
2
+
γ˜ 2 ρ 2 4
+ iγ Lˆ z ;
(1)
/ ,
4V0 R20
which describes the on-axis D0 and the electron confined in a semiconductor heterostructure with axial symmetry for Q = 1 and Q = 0, respectively: Here lengths are scaled in the units of the Bohr effective radius a∗0 = ε h¯ 2 /m∗ e2 and the energy in the effective Rydberg Ry∗ = e2 /2ε a∗0 . Parameters α = ea∗0 F /Ry∗ and γ = eh¯ B/2m∗ cRy∗ are dimensionless units of the strengths of the applied electric, F , and magnetic, B, fields along the symmetry axis. The electron and the on-axis donor position vectors are given by the cylindrical coordinates (ρ, z , φ) and (0, ξ , 0), respectively. For the sake of the mathematical convenience we consider in this work the separable axially symmetrical confinement potential Vc (ρ, z ) = V (z ) + V0 ρ 2 /R20 with parabolic confinement in the radial direction in which parameters R0 and V0 are associated with the nanostructure radius and the barrier height in the lateral junction, respectively. In the large R0 limit, this potential is converted into corresponding potentials of the QW, MQW or SL depending on the choice of the potential V (z ), which describes the confinement
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in the axial direction. The last term in the Hamiltonian (1) takes into account the interaction of the angular momentum with the magnetic field. The resulting potential in donors confined in the heterostructures is no longer centrally symmetric. Therefore there is no law of conservation of the total angular momentum and it is impossible to classify the electronic states according to the values of the total orbital angular momentum L. But if the structure has axial symmetry, then the projection of the orbital angular momentum on this axis is conserved, and we can classify the electron states of donors confined in such structures according to the values of this projection. The absolute value of the projected orbital angular momentum along the axis could be denoted similarly to the axially symmetrical molecules by the letter Λ; it takes the values 0, 1, 2, . . .. Furthermore, the donor electronic states with different values of Λ in this case could be denoted similarly to the molecular states by the capital Greek letters corresponding to the Latin letters for atomic states with different Λ. Thus, for the states with Λ = 0, 1, 2 we use notations Σ , Π and ∆, respectively. All electron states with non-zero values of Λ are doubly degenerated: to each value of the energy there correspond two states which differ in the direction of the projection of the orbital angular momentum on the symmetry axis. The Σ states are also doubly degenerated, one state, denoted by Σ + , is unaltered and the other state, denoted by Σ − , has a wave function that changes sign on reflection in a plane of symmetry passing through the axis. The electron terms of the donor are described by the Hamiltonian HΛ , defined as (0)
HΛ = HΛ (Q = 1) + V (z ) + α z + γ · Λ; Λ = 0, ±1, ±2, . . . 2Q γ˜ 2 ρ 2 1 ∂ ∂ ∂2 Λ2 (0) HΛ (Q ) = −∆Λ − + ; ∆Λ = ρ + 2 − 2; r 4 ρ ∂ρ ∂ρ ∂z ρ q r =
(2)
ρ 2 + (z − ξ )2 .
(0)
The singularity of the Coulomb potential in HΛ at short distances r from the donor places additional constraints on the wave functions of the Hamiltonian (2). If the electron approaches the donor then the (0) potential term in HΛ becomes very large making depreciable the influence of the confinement and the external fields. As the result the spherical symmetry is restored and the behavior of the wavefunctions becomes similar to those of a free donor. A question thus arises about the correspondence between electronic states of the confined and the unconfined donors obtained by comparing the behavior of the wave functions at short and at large electron–ion separations. At short distances between the electron and the donor, when the potential can be considered as almost spherically symmetrical, pure states, which are significantly degenerated, are given by three quantum numbers, principal n, orbital l and azimuthal m (denoted in what follows as Λ). The corresponding orbitals are χn,l,Λ (ρ, z ) = Rn,l (r )Pl,Λ ((z − ξ ) /r ), where Rn,l (r ) and Pl,m ((z − ξ ) /r ) are the respective radial and angular parts of the hydrogen wave functions corresponding to these orbitals centred at the donor location. The singularity of the Coulomb potential in the vicinity of the donor position may be canceled by a corresponding divergence in the kinetic energy term. Taking into account that inside an infinitesimal spherical region around the D0 the eigenfunctions of the Hamiltonian (2) are similar to the hydrogen atom orbitals, which are characterized by three quantum numbers n, l, Λ, one can demonstrate that two terms are canceled only if the wave function have a ‘‘cusp’’ at r = ξ , which means that they should satisfy
|r − ξ |l d lim r→ξ ψ(r) dr
ψ(r) 1 =− ; l l+1 |r − ξ |
l = 0, 1, 2, . . . .
(3)
Here l is orbital quantum number, which at short distances from the D0 again becomes a ‘‘good’’ quantum number in spite of the fact that the confinement potential in the heterostructure is anisotropic. In this sense, different sublevels of the donor, corresponding to different values of a single ‘‘good’’ quantum number Λ, can be referred as s-like states Σ (s) , Π (s) , ∆(s) , . . . if they satisfy the ‘‘cusp condition (3) with l = 0, p-like states Σ (p) , Π (p) , ∆(p) , . . ., if they satisfy the condition (3) with l = 1, etc., i.e. in concordance with their symmetries at short distances from the D0 . Here the capital Greek labels are true quantum numbers that give the angular momentum projection on the symmetry axis, whereas the superscripts show the similarity of the sublevels’ wavefunctions to
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the correspondent pure hydrogenic orbitals at short distances. On withdrawing from the donor this similarity becomes gradually weaker due to the stronger mixing of the hydrogenic states under the increasing influence of the structural confinement and the external fields. The larger the separation between the electron and the donor the stronger is the mixing of the hydrogenic states and the closer is the correspondence between the long-range symmetry of the resulting donor wave functions and the symmetry of the heterostructure. This condition can be satisfied by the introduction in the wave function of an additional factor that has the same symmetry as the confinement potential. Following this idea we represent the eigenfunctions of the Hamiltonian (2) in the form:
ˆ Λ Ψn,l,Λ = En,l,Λ Ψn,l,Λ ; H
Ψn,l,Λ = χn,l,Λ (ρ, z ) fΛ (ρ)Φn,l,Λ (z );
2 fΛ (ρ) = ρ |Λ| e−γ ρ /4 .
(4)
Here Φn,l,Λ (z ) and fΛ (ρ) are envelope functions that describe the transformation of Slater orbitals due to confinement along and across the symmetry axis, respectively. In order to derive the differential equation for the unknown function Φn,l,Λ (z ) we consider the functional: F Φn,l,Λ = fΛ (ρ)χn,l,Λ (ρ, z ) Φn,l,Λ (z ) |HΛ − E | fΛ (ρ) χn,l,Λ (ρ, z ) Φn,l,Λ (z ) .
(5)
In accordance with the Schrödinger variational principle the best of all trial functions (4) is given by the function Φn,l,Λ (z ) for which the functional F Φn,l,Λ has a stationary value. Substituting the Hamiltonian (2) in the functional (5) one can find the explicit expression for the functional (5). The variational principle then reduces to the differential equation:
h(z ) + V (z ) + α z + Φn,l,Λ (z ) J (z ) dz dz 2J (z ) = En,l,Λ − EΛ − En − γ Λ Φn,l,Λ (z ).
−
1
d
J (z )
dΦn,l,Λ (z )
(6)
Here En = −1/n2 ; J (z ) =
0
h(z ) =
EΛ = γ˜ (|Λ| + 1) ;
(6a)
fΛ2 (ρ)χn2,l,Λ (ρ, z ) ρ dρ = BΛ (z );
(6b)
∞
Z
∞
Z
χ
0
2 n,l,Λ
(ρ, z )
d dρ
(
d fΛ2 (ρ)
ρ
dρ
) dρ = 4Λ2 BΛ−1 (z ) + γ˜ 2 BΛ+1 (z )
− 2γ˜ (2 |Λ| + 1) BΛ (z ); (6c) q Z ∞ p 2 2 BΛ (z ) = ρ 2|Λ|+1 e−γ˜ ρ /2 R2n,l ρ 2 + (z − ξ )2 · Yl2,Λ (z − ξ ) / ρ 2 + (z − ξ ) dρ. (6d) 0
In addition, the differential equation (6) should be complemented by two frontier conditions, the large-range and short-range. For localized donor states, the electron is situated mainly in the vicinity of the donor location (ξz − Rmax < z < ξz + Rmax ). The behavior of the envelope function far from the donor location is not important because of the rapid decrease of the factor corresponding to the Slater orbital on retreating from the donor. Therefore we assume in our calculations the large-range frontier condition Φ (ξz + Rmax ) = 0, where we choose the value Rmax equal to a few effective Bohr radii. On the other hand, the value of the envelope function close to the donors is significant but the ‘‘cusp’’ condition (3) is satisfied only if the short-range frontier condition Φ 0 (ξz ) = 0 is fulfilled. Thus Eqs. (6) together with these two frontier conditions present the boundary value problem which we solve by using the trigonometric sweep method [9]. The attractive aspect of our method of calculation of the donor energy spectrum consists in its universality and applicability to any heterostructure that has axial symmetry. Once the energies En,l,Λ are found, the binding energies Eb (n, l, Λ) then may be found by subtracting these values from the lowest free electron energy. In our numerical work we have found Eb (n, l, Λ) as differences between the results of two calculations, obtained by first putting in the above formulae Q = 0, n = 0, l = 0, Λ = 0, and then putting Q = 1.
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3. Results and discussions In order to check the accuracy of our procedure we first compute the binding energies corresponding to the ground Σ (1s) state for on-center donor GaAs/Ga0.75 Al0.25 As QW with well widths 10 and 20 nm. Our results are posted in Table 1 along with the diffusion quantum Monte Carlo results [3] for various magnetic fields. It is clear that our method yields binding energies that are in an excellent concordance with the Monte Carlo calculation. Thus our trial function is reasonable and it provides sufficiently accurate results for the D0 ground state energy in QW. Table 1 Comparison of D0 binding energies for the ground Σ (1s) state with Monte Carlo results from Ref. [3].
γ
L (nm)
This work
Ref. [3]
0 0 1 1 3 3
10 20 10 20 10 20
2.053 1.733 2.942 2.497 2.899 3.290
2.090 ± 0.040 1.740 ± 0.030 2.920 ± 0.060 2.520 ± 0.050 2.890 ± 0.050 3.360 ± 0.070
In Fig. 1(a, b, c) we compare our results for the ground state energies of donors confined in single and multiple heterostructures with those obtained previously by using other calculation methods. The ground state binding energy of the donor, placed at the center of a single GaAs/Ga0.75 Al0.25 As QW, as a function of the QW width is plotted in Fig. 1(a) together with the results obtained previously by using the variational [10], fractal-dimensional [11] and Monte Carlo [3] methods. It is seen that in general the agreement between our results and all other methods is excellent considering the relative simplicity of our procedure. Nevertheless, there are small discrepancies between our results and those obtained by means of variational techniques when the well is very narrow or very wide, the concordance being better between our and the Monte Carlo method for intermediate values of the QW width. We believe that our trial function is more flexible and it adjusts better to the symmetry of the heterostructure in comparison to other trial functions. In Fig. 1(b) we compare our results (solid lines) with calculations realized in Ref. [12] by using the finite difference method (symbols) for the dependencies of the ground state energies on the width of the barriers between the wells in a GaAs/Ga0 .67Al0.33 As MQW system consisting of five 15 nm QWs, for an impurity placed at the center of the central well. The good agreement with more sophisticated techniques such as the finite difference method once again confirms the validity of the present more simple treatment. Also we compare our calculations for donors in a triple GaAs/Ga0.7 Al0.3 As QW with those obtained in Ref. [13] by means of the variational method, where a linear combination of Gaussian functions with at least 300 variational parameters has been used as a trial function to guarantee an adequate convergence. In Ref. [13] the authors considered a structure formed by a 40 Å central well coupled through 20 Å thin barriers to lateral wells of 140 Å on both sides. The barriers heights for the lateral and central wells in the considered structure were taken as 200 and 50 meV, respectively. In Fig. 1(c) we present binding energies in such a structure calculated as functions of the impurity position for three different values of the intensity of magnetic field. The agreement between our results (solid lines) and those from Ref. [13] (symbols) is excellent. In Fig. 2 we plot the binding energies of three first s-like excited states as functions of the QW width for a donor placed at the center of the well. It is seen that the donor binding energies for large well widths descend slowly towards the corresponding well-known bulk exact values, i.e., to Ry∗ /4 for the Σ (2s) state, to Ry∗ /9 for the Σ (3s) state, and to Ry∗ /16 for the Σ (4s) state, respectively. On the opposite sides of the curves, as the well width decreases, the binding energies initially increase slightly but then drop abruptly tending to zero as the well width approaches 4a∗0 , i.e. when the distance from the donor to the barrier is equal to 2a∗0 . We associate this result with the impossibility of forming donor excited states, stable with respect to the ionization, when the first node in the z-direction for the wave functions is situated outside the
W. Gutiérrez et al. / Superlattices and Microstructures 48 (2010) 288–297
a
293
b
. . . . . .
c
Fig. 1. Comparison of our calculation results for the ground state donor energies confined in different GaAs/Ga(Al)As heterostructures with those obtained previously by other methods: (a) the on-center donor binding energy as a function of a single QW width; (b) D0 energy as function of width of the barriers between the wells in a MQW system consisting of five 15 nm QWs for an impurity placed at the center of the central well; (c) D0 binding energies in a triple QW as a function of the impurity position for different values of the magnetic field. .
.
.
.
Fig. 2. On-center donor binding energies of three s-like excited states as a function of the GaAs/Ga0.3 Al0.7 As QW width. and (b) the applied electric field, for three donor positions.
well. It is worth mentioning that the hydrogen wave functions corresponding to the three excited states, whose curves are presented in Fig. 2, have the first node in the radial direction close the point r = 2a∗0 . In what follows we present results of the analysis of the effect of the electric field on the ground state of an on-axis donor in GaAs/Ga1−x Alx As SL and NWSL with a cylindrical cross-section of radius R and with variable concentration of Al along the crystal growth z direction, which is equal to zero inside the n identical wells of width a and to x in the n − 1 barriers of width b. We assume that the
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a
b
c
Fig. 3. Ground state energy of an on-axis donor in heterostructures with 11 unit cells (a = 10 nm, b = 5 nm) as a function of the impurity position, under different electric fields in (a) SL, and in NWSL of radii (b) R = 40 nm and (c) R = 10 nm.
confinement potential in such a structure in the z direction is given by a piecewise constant function V (z ), which is equal to zero in the wells and to V0 in the barriers. In Figs. 3–5 we present the results of our calculations for the GaAs/Ga0.7 Al0.3 As SL with a = 10 nm, b = 5 nm and for NWSL with the same parameters a and b and two different radii: 10, 40 nm. The effective Rydberg, Ry∗ , in these structures is about 5.83 meV, the Bohr radius, a∗0 , about 10 nm and the barrier height, V0 , about 40 Ry*. Dependencies of the donor energies on the distance ξ from the left end of the NWSL for several values of the external electric field are shown in Fig. 3. It is seen that for zero electric field the donor energy variation with the displacement of the donor position from the middle of the heterostructures towards one of the ends of the NWSL is wavy. This undulating behavior of the energy dependence is caused by the variation of the probability of finding the electron in different parts of the NWSL. The higher the probability to find the electron at the donor location, the stronger is the attraction between the electron and the ion, and the greater is the drop in energy. When electric field is absent, the greatest probability of finding the electron is at the middle of the central well and therefore at this point the energy is minimal. When the donor is moved from this point towards one of the NWSL ends the donor energy E D0 is increased due to the increasing average separation between the electron
and the ion, until E D0 reaches a local maximum at the center of the first adjacent barrier, where the probability to find the electron is smaller than in the well. As the donor is moved further toward the NWSL end, it enters the neighboring well, and the energy is decreased again until the donor reaches the center of this well. Since the density in the neighboring well is smaller than in the central well, the new local minimum is higher than main minimum in the central well. Also, one can observe in Fig. 3 that the energy dependence is altered essentially in the presence of the external electric field. The undulating character of the ground state energy dependence disappears and the position of the main energy minimum is displaced sharply to the center of the first well. Comparing Fig. 3(a)–(c) one can see that as the NWSL radius decreases a stronger electric field should be applied in order to remove the undulating energy dependencies.
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a
295
b
. . . . . . .
c
. . . . . .
Fig. 4. Ground state energy of a donor located at the center of the central well in the GaAs/Ga0.7 Al0.3 As (a) SL, and NWSL of radii (b) R = 40 nm and (c) R = 10 nm with different numbers of wells as a function of the electric field.
There is a strong correlation between the probability density function of the electron in the heterostructure and the donor binding energy. Basically, a higher the probability of finding the electron at the donor location leads to the electron spending more time close to the donor, therefore resulting in a higher binding energy. The greater the radius of the NWSL or the number of wells, the weaker is the localization of the electron inside the wells and the easier it is for the electron to be displaced far from the donor location under an electric field. In order to reveal this fact we present in Fig. 4 the calculation results for the binding energies of donors located at the center of the central well as functions of the electric field strength in SL and two NWSLs of radii 40 and 10 nm and with one, three, five, and eleven wells. As the electric field increases, the electron is pulled away from the ion toward one of the ends of the heterostructure, the electron wave function becomes more extended, and the donor binding energy drops, due to an increase in the average distance between the electron and ion, until it reaches a minimum, when the electron is mainly located close to one of the ends of the heterostructure. One can see in Fig. 4 that both the slope of the initial part of curves and the minimum value of the binding energy depend on the number of wells and the radius of the NWSL. The greater the number of the wells, the longer the wire, the stronger the lowering of the potential energy due to the electric field, and the easier it is to pull the electron away from the ion, resulting in an increase of the slope in the curve of the binding energy dependence on the electric field. A similar change of the slope of the curves can be observed in Fig. 4 with an increase of the radius of the NWSL. An increase of the NWSL radius provides a reduction of the electron confinement inside the wells and leads to a stronger tunneling of the electron along the axis under an external electric field applied along the axis, resulting in an increase of the slope of the curves, as observed in Fig. 4(a)–(c). Dependencies of the binding energies on the electric field for an impurity located at the center of the first well of the NWSL are shown in Fig. 5. In this case the electron is pulled toward the donor under an external electric field and therefore the binding energy increases when the electric field is increased. In contrast to the dependencies presented in Fig. 4, the binding energies in Fig. 5 in a strong electric field tend to the same value for different numbers of the wells in the heterostructure. This is due to the fact that the electron under a strong electric field is pulled inside the same cell where the donor is located and therefore the average separation between the electron and ion does not depend on the number of wells.
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a
b .
. . .
.
. .
.
.
c
. . . . . .
Fig. 5. Ground state energy of a donor located at the center of the first well in the GaAs/Ga0.7 Al0.3 As (a) SL, and NWSL of radii (b) R = 40 nm and (c) R = 10 nm with different numbers of wells as a function of the electric field.
4. Summary and conclusions In this work we have proposed a new nomenclature for different electronic states of a donor confined in a semiconductor heterostructure with axial symmetry, and have presented a new simple method to calculate the corresponding wave functions and energies, based on the similarity of the ‘‘cusp’’ conditions for a donor confined in a heterostructure and those for a hydrogenic atom. The energy levels in our nomenclature are labeled as Λ(nl) , characterized by one ‘‘good’’ quantum number Λ, the projection of the orbital angular momentum on the symmetry axis, and two quantum numbers, n and l, which are considered as ‘‘good’’ only inside a restricted region close to the donor position, where the influence of the confinement is negligible. The donor eigenfunctions are represented as a product of the corresponding hydrogenic atom orbitals with an envelope function which takes into account the modification of these orbitals due to the confinement along and across the symmetry axis. Starting from the Schrödinger variational principle we derive an universal differential equation for this envelope function, applicable to any heterostructure with axial symmetry. The method has been used to calculate the ground and some excited energy states of donors confined in single and multiple quantum wells, and nanowire-superlattices for different magnetic and electric fields. The accuracy of our procedure is verified by the good accordance between our results for a single and multiple quantum wells with those previously obtained by using the Monte Carlo, finite difference and variational methods. One of the attractive features of our method is its universality and the simplicity of its application to different models of confinement in the framework of the same numerical procedure. It permits one to avoid tedious calculations in order to obtain, with a high accuracy, the donor energies in any heterostructure once the confinement potential is known. We present new results of application of the method to cylindrical GaAs–(Ga,Al)As nanowiresuperlattices in order to analyze the effect of the variation of the number of wells, the wire radius, the donor position, and the electric field strength. Similarly, problems for excitons or for other carrier complexes, confined in heterostructures with axial symmetry can be treated. Thus, our method can be applied to study the properties of different bound carrier states (neutral and negatively charged
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donors, acceptors, excitons, biexcitons, trions and so on) confined in different types of heterostructure in the presence of any external perturbation, and it may even exhibit practical advantages, similar to those shown in our work. Acknowledgements This work was financed by the Industrial University of Santander (UIS) through the Dirección General de Investigaciones (DIF Ciencias, Cod. 5111) and the Excellence Center of Novel Materials— ECNM, under Contract No. 043-2005 and the Cod. No. 1102-05-16923 subscribed with COLCIENCIAS. References [1] M. Grundmann (Ed.), Nano-Optoelectronics: Concepts, Physics and Devices, Springer-Verlag, Berlin, 2002. [2] G. Bastard, Phys. Rev. B24 (1981) 4714; V. Branis, G. Li, K.K. Bajaj, Phys. Rev. B 47 (1993) 1316; S.T. Pérez-Merchancano, G.E. Marques, Solid State Commun. 110 (1999) 209. [3] T. Pang, S.G. Louie, Phys. Rev. Lett. 65 (1990) 1635. [4] S. Glutsch, Excitons in Low-Dimensional Semiconductors, Springer Verlag, 2004. [5] J.L. Zhu, J.H. Zhao, J.J. Xiong, Phys. Rev. B 50 (1994) 1832; A. Gonzalez, I. Mikhailov, Int. J. Mod. Phys. B 11 (1997) 3469; M. El-Said, Physica B 202 (1994) 202. [6] R.L. Greene, K.K. Bajaj, Solid State Commun. 45 (1983) 825; Phys. Rev. B 31 (1985) 913; A. Latge, N. Porras-Montenegro, L.E. Oliveira, Phys. Rev B. 45 (1992) 9420. [7] S. Chaudhuri, K.K. Bajaj, Phys. Rev. B 29 (1984) 1803; H. Sari, E. Kasapoglu, I. Sokmen, Physica B 325 (2003) 300. [8] M. Helm, W. Hilber, T. Fromherz, F.M. Peeters, K. Alavi, R.N. Pathak, Phys. Rev. B 48 (1993) 1601; M. Helm, F.M. Peeters, F. DeRosa, E. Colas, J.P. Harbison, L.T. Florez, Phys. Rev. B 43 (1991) 13983; J.M. Shi, F.M. Peeters, G.Q. Hai, J.T. Devreese, Phys. Rev. B 44 (1991) 5692; J.M. Shi, F.M. Peeters, G.Q. Hai, J.T. Devreese, Phys. Rev. B 50 (1994) 15182. [9] F.J. Betancur, I.D. Mikhailov, L.E. Oliveira, J. Phys. D 31 (1998) 3391. [10] S. Fraizzoli, F. Bassani, R. Buczko, Phys. Rev. B 41 (1990) 5096. [11] I.D. Mikhailov, F.J. Betancur, R.A. Escorcia, J. Sierra-Ortega, Phys. Rev. B 67 (2003) 115317. [12] T. Kuhn, G. Mahler, J.L. Dunn, C.A. Bates, J. Phys.: Condens. Matter 6 (1994) 757. [13] M. Pacheco, Z. Barticevic, A. Latgé, Physica B 302 (2001) 77.
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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Localized donor states in axially symmetrical heterostructures W. Gutiérrez ∗ , L.F. García, I.D. Mikhailov Escuela de Física, Universidad Industrial de Santander, A.A. 678, Bucaramanga, Colombia
article
info
Article history: Received 19 February 2010 Received in revised form 13 May 2010 Accepted 17 June 2010 Available online 18 July 2010 Keywords: Semiconductors nanostructures Neutral donor Binding energy Excited states Nanowire superlattice
abstract We present a general method for calculating the energy spectrum of donors confined in heterostructures with axial symmetry in the presence of magnetic and electric fields applied along the symmetry axis. The donor’s wave functions are chosen as a product of the Slater orbitals and an envelope function that is a solution of a one-dimensional differential equation, which we derive starting from Schrödinger’s variational principle. We calculate the energies of the ground and some excited states of a donor confined in multiple quantum wells and a nanowire superlattice as functions of the donor position, electric and magnetic fields for structures with different numbers and widths of the wells and the barriers. Our method could be applicable to a variety of complex quantum-confined semiconductor structures for which more rigorous approaches require extensive numerical calculations. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction For the past few decades different quasi-two-dimensional semiconductor structures, such as single and double quantum wells (DQWs), multiple quantum wells (MQWs), superlattices (SLs) and nanowire superlattices (NWSLs), have been studied extensively both theoretically and experimentally, and applied to various electronic and photonic devices. Taking electric and magnetic fields applied along to the growth direction together with the effects of the confinement and tunneling across the potential barriers, we have an interesting physical system in which the competition of these factors allows us to manipulate its electro-optical properties conveniently for applications in spintronics, optoelectronics, photovoltaics, and quantum information technologies [1].
∗
Corresponding author. Tel.: +57 7 6323095; fax: +57 7 6332477. E-mail address: [email protected] (W. Gutiérrez).
0749-6036/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2010.06.016
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The widespread interest in semiconductor structures with axial symmetry is related to the ability to manipulate the energy splitting between the lowest-lying states or the spatial distribution of carriers by varying the radius (in the case of the NWSL), thicknesses and number of wells and barriers. The electrical and optical properties of these structures also may be modified by impurities due to alteration of the carriers’ spatial distribution, to shift and to splitting of the low-lying energy levels in their presence. The resulting alteration of the electro-optical properties of the heterostructures depends on various factors, such as the donor position, the quantum confinement, the interwell tunneling and the electron–ion attraction. For this reason the elaboration of a simple and accurate method of calculation of the energy spectrum of donors (D0 ) confined in different types of the heterostructures has been the subject of intense research since the advent of nanotechnology. The variational [2], Monte Carlo [3], finite difference [4] and other methods [5] have been used for calculating the low lying energy levels of D0 confined in various types of heterostructures, such as QW [6], MQW [7], QWW and SL [8]. Nevertheless, the elaboration of a unified approach to the theoretical analysis of the localized donor states common to all heterostructures has been difficult due to the additional constraints that are imposed on the electron wave function at short distances by the singularity of the Coulomb potential. If the electron approaches the ion then the potential becomes large and negative. This must be canceled by a corresponding positive divergence in the kinetic energy. This condition can be satisfied only if the corresponding wave functions have a ‘‘cusp’’ at the point of the donor location, identical for all heterostructures independently of their geometries. Any accurate algorithm should take into account this circumstance in order to ensure a rapid convergence of the estimated D0 energies to their exact values. In this work a particular emphasis is given to the choice of the optimal wave functions for localized donor states which satisfy the ‘‘cusp’’ condition. We were motivated to analyze this subject by the interest in the elaboration a universal technique for calculating the energies and wave functions of the localized states in nanostructures induced by the presence in them of a shallow donor. Our main result is a general one-dimensional differential equation for the envelope function derived in Section 2, which allows us to analyze, in a simple form, modifications of the hydrogenic states due to the confinement in any axially symmetrical heterostructure. In Section 3 we compare results obtained with our method for ground state donor energies in different types of heterostructures, such as QW, MQW and SL, with corresponding results obtained previously by using other, more sophisticated techniques. We analyze dependencies of the donor binding energy in these structures on the geometric parameters, donor position, and intensities of the electric and the magnetic fields. Also, we present novel results for some donor s-like states in QW and NWSL. 2. Theory The object of our analysis is the dimensionless Hamiltonian H = H (0) (Q ) + V (z ) + α z ;
γ˜ = γ + 2
2
H (0) (Q ) = −∆ − p
2Q
ρ 2 + (z − ξ )
2
+
γ˜ 2 ρ 2 4
+ iγ Lˆ z ;
(1)
/ ,
4V0 R20
which describes the on-axis D0 and the electron confined in a semiconductor heterostructure with axial symmetry for Q = 1 and Q = 0, respectively: Here lengths are scaled in the units of the Bohr effective radius a∗0 = ε h¯ 2 /m∗ e2 and the energy in the effective Rydberg Ry∗ = e2 /2ε a∗0 . Parameters α = ea∗0 F /Ry∗ and γ = eh¯ B/2m∗ cRy∗ are dimensionless units of the strengths of the applied electric, F , and magnetic, B, fields along the symmetry axis. The electron and the on-axis donor position vectors are given by the cylindrical coordinates (ρ, z , φ) and (0, ξ , 0), respectively. For the sake of the mathematical convenience we consider in this work the separable axially symmetrical confinement potential Vc (ρ, z ) = V (z ) + V0 ρ 2 /R20 with parabolic confinement in the radial direction in which parameters R0 and V0 are associated with the nanostructure radius and the barrier height in the lateral junction, respectively. In the large R0 limit, this potential is converted into corresponding potentials of the QW, MQW or SL depending on the choice of the potential V (z ), which describes the confinement
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in the axial direction. The last term in the Hamiltonian (1) takes into account the interaction of the angular momentum with the magnetic field. The resulting potential in donors confined in the heterostructures is no longer centrally symmetric. Therefore there is no law of conservation of the total angular momentum and it is impossible to classify the electronic states according to the values of the total orbital angular momentum L. But if the structure has axial symmetry, then the projection of the orbital angular momentum on this axis is conserved, and we can classify the electron states of donors confined in such structures according to the values of this projection. The absolute value of the projected orbital angular momentum along the axis could be denoted similarly to the axially symmetrical molecules by the letter Λ; it takes the values 0, 1, 2, . . .. Furthermore, the donor electronic states with different values of Λ in this case could be denoted similarly to the molecular states by the capital Greek letters corresponding to the Latin letters for atomic states with different Λ. Thus, for the states with Λ = 0, 1, 2 we use notations Σ , Π and ∆, respectively. All electron states with non-zero values of Λ are doubly degenerated: to each value of the energy there correspond two states which differ in the direction of the projection of the orbital angular momentum on the symmetry axis. The Σ states are also doubly degenerated, one state, denoted by Σ + , is unaltered and the other state, denoted by Σ − , has a wave function that changes sign on reflection in a plane of symmetry passing through the axis. The electron terms of the donor are described by the Hamiltonian HΛ , defined as (0)
HΛ = HΛ (Q = 1) + V (z ) + α z + γ · Λ; Λ = 0, ±1, ±2, . . . 2Q γ˜ 2 ρ 2 1 ∂ ∂ ∂2 Λ2 (0) HΛ (Q ) = −∆Λ − + ; ∆Λ = ρ + 2 − 2; r 4 ρ ∂ρ ∂ρ ∂z ρ q r =
(2)
ρ 2 + (z − ξ )2 .
(0)
The singularity of the Coulomb potential in HΛ at short distances r from the donor places additional constraints on the wave functions of the Hamiltonian (2). If the electron approaches the donor then the (0) potential term in HΛ becomes very large making depreciable the influence of the confinement and the external fields. As the result the spherical symmetry is restored and the behavior of the wavefunctions becomes similar to those of a free donor. A question thus arises about the correspondence between electronic states of the confined and the unconfined donors obtained by comparing the behavior of the wave functions at short and at large electron–ion separations. At short distances between the electron and the donor, when the potential can be considered as almost spherically symmetrical, pure states, which are significantly degenerated, are given by three quantum numbers, principal n, orbital l and azimuthal m (denoted in what follows as Λ). The corresponding orbitals are χn,l,Λ (ρ, z ) = Rn,l (r )Pl,Λ ((z − ξ ) /r ), where Rn,l (r ) and Pl,m ((z − ξ ) /r ) are the respective radial and angular parts of the hydrogen wave functions corresponding to these orbitals centred at the donor location. The singularity of the Coulomb potential in the vicinity of the donor position may be canceled by a corresponding divergence in the kinetic energy term. Taking into account that inside an infinitesimal spherical region around the D0 the eigenfunctions of the Hamiltonian (2) are similar to the hydrogen atom orbitals, which are characterized by three quantum numbers n, l, Λ, one can demonstrate that two terms are canceled only if the wave function have a ‘‘cusp’’ at r = ξ , which means that they should satisfy
|r − ξ |l d lim r→ξ ψ(r) dr
ψ(r) 1 =− ; l l+1 |r − ξ |
l = 0, 1, 2, . . . .
(3)
Here l is orbital quantum number, which at short distances from the D0 again becomes a ‘‘good’’ quantum number in spite of the fact that the confinement potential in the heterostructure is anisotropic. In this sense, different sublevels of the donor, corresponding to different values of a single ‘‘good’’ quantum number Λ, can be referred as s-like states Σ (s) , Π (s) , ∆(s) , . . . if they satisfy the ‘‘cusp condition (3) with l = 0, p-like states Σ (p) , Π (p) , ∆(p) , . . ., if they satisfy the condition (3) with l = 1, etc., i.e. in concordance with their symmetries at short distances from the D0 . Here the capital Greek labels are true quantum numbers that give the angular momentum projection on the symmetry axis, whereas the superscripts show the similarity of the sublevels’ wavefunctions to
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the correspondent pure hydrogenic orbitals at short distances. On withdrawing from the donor this similarity becomes gradually weaker due to the stronger mixing of the hydrogenic states under the increasing influence of the structural confinement and the external fields. The larger the separation between the electron and the donor the stronger is the mixing of the hydrogenic states and the closer is the correspondence between the long-range symmetry of the resulting donor wave functions and the symmetry of the heterostructure. This condition can be satisfied by the introduction in the wave function of an additional factor that has the same symmetry as the confinement potential. Following this idea we represent the eigenfunctions of the Hamiltonian (2) in the form:
ˆ Λ Ψn,l,Λ = En,l,Λ Ψn,l,Λ ; H
Ψn,l,Λ = χn,l,Λ (ρ, z ) fΛ (ρ)Φn,l,Λ (z );
2 fΛ (ρ) = ρ |Λ| e−γ ρ /4 .
(4)
Here Φn,l,Λ (z ) and fΛ (ρ) are envelope functions that describe the transformation of Slater orbitals due to confinement along and across the symmetry axis, respectively. In order to derive the differential equation for the unknown function Φn,l,Λ (z ) we consider the functional: F Φn,l,Λ = fΛ (ρ)χn,l,Λ (ρ, z ) Φn,l,Λ (z ) |HΛ − E | fΛ (ρ) χn,l,Λ (ρ, z ) Φn,l,Λ (z ) .
(5)
In accordance with the Schrödinger variational principle the best of all trial functions (4) is given by the function Φn,l,Λ (z ) for which the functional F Φn,l,Λ has a stationary value. Substituting the Hamiltonian (2) in the functional (5) one can find the explicit expression for the functional (5). The variational principle then reduces to the differential equation:
h(z ) + V (z ) + α z + Φn,l,Λ (z ) J (z ) dz dz 2J (z ) = En,l,Λ − EΛ − En − γ Λ Φn,l,Λ (z ).
−
1
d
J (z )
dΦn,l,Λ (z )
(6)
Here En = −1/n2 ; J (z ) =
0
h(z ) =
EΛ = γ˜ (|Λ| + 1) ;
(6a)
fΛ2 (ρ)χn2,l,Λ (ρ, z ) ρ dρ = BΛ (z );
(6b)
∞
Z
∞
Z
χ
0
2 n,l,Λ
(ρ, z )
d dρ
(
d fΛ2 (ρ)
ρ
dρ
) dρ = 4Λ2 BΛ−1 (z ) + γ˜ 2 BΛ+1 (z )
− 2γ˜ (2 |Λ| + 1) BΛ (z ); (6c) q Z ∞ p 2 2 BΛ (z ) = ρ 2|Λ|+1 e−γ˜ ρ /2 R2n,l ρ 2 + (z − ξ )2 · Yl2,Λ (z − ξ ) / ρ 2 + (z − ξ ) dρ. (6d) 0
In addition, the differential equation (6) should be complemented by two frontier conditions, the large-range and short-range. For localized donor states, the electron is situated mainly in the vicinity of the donor location (ξz − Rmax < z < ξz + Rmax ). The behavior of the envelope function far from the donor location is not important because of the rapid decrease of the factor corresponding to the Slater orbital on retreating from the donor. Therefore we assume in our calculations the large-range frontier condition Φ (ξz + Rmax ) = 0, where we choose the value Rmax equal to a few effective Bohr radii. On the other hand, the value of the envelope function close to the donors is significant but the ‘‘cusp’’ condition (3) is satisfied only if the short-range frontier condition Φ 0 (ξz ) = 0 is fulfilled. Thus Eqs. (6) together with these two frontier conditions present the boundary value problem which we solve by using the trigonometric sweep method [9]. The attractive aspect of our method of calculation of the donor energy spectrum consists in its universality and applicability to any heterostructure that has axial symmetry. Once the energies En,l,Λ are found, the binding energies Eb (n, l, Λ) then may be found by subtracting these values from the lowest free electron energy. In our numerical work we have found Eb (n, l, Λ) as differences between the results of two calculations, obtained by first putting in the above formulae Q = 0, n = 0, l = 0, Λ = 0, and then putting Q = 1.
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3. Results and discussions In order to check the accuracy of our procedure we first compute the binding energies corresponding to the ground Σ (1s) state for on-center donor GaAs/Ga0.75 Al0.25 As QW with well widths 10 and 20 nm. Our results are posted in Table 1 along with the diffusion quantum Monte Carlo results [3] for various magnetic fields. It is clear that our method yields binding energies that are in an excellent concordance with the Monte Carlo calculation. Thus our trial function is reasonable and it provides sufficiently accurate results for the D0 ground state energy in QW. Table 1 Comparison of D0 binding energies for the ground Σ (1s) state with Monte Carlo results from Ref. [3].
γ
L (nm)
This work
Ref. [3]
0 0 1 1 3 3
10 20 10 20 10 20
2.053 1.733 2.942 2.497 2.899 3.290
2.090 ± 0.040 1.740 ± 0.030 2.920 ± 0.060 2.520 ± 0.050 2.890 ± 0.050 3.360 ± 0.070
In Fig. 1(a, b, c) we compare our results for the ground state energies of donors confined in single and multiple heterostructures with those obtained previously by using other calculation methods. The ground state binding energy of the donor, placed at the center of a single GaAs/Ga0.75 Al0.25 As QW, as a function of the QW width is plotted in Fig. 1(a) together with the results obtained previously by using the variational [10], fractal-dimensional [11] and Monte Carlo [3] methods. It is seen that in general the agreement between our results and all other methods is excellent considering the relative simplicity of our procedure. Nevertheless, there are small discrepancies between our results and those obtained by means of variational techniques when the well is very narrow or very wide, the concordance being better between our and the Monte Carlo method for intermediate values of the QW width. We believe that our trial function is more flexible and it adjusts better to the symmetry of the heterostructure in comparison to other trial functions. In Fig. 1(b) we compare our results (solid lines) with calculations realized in Ref. [12] by using the finite difference method (symbols) for the dependencies of the ground state energies on the width of the barriers between the wells in a GaAs/Ga0 .67Al0.33 As MQW system consisting of five 15 nm QWs, for an impurity placed at the center of the central well. The good agreement with more sophisticated techniques such as the finite difference method once again confirms the validity of the present more simple treatment. Also we compare our calculations for donors in a triple GaAs/Ga0.7 Al0.3 As QW with those obtained in Ref. [13] by means of the variational method, where a linear combination of Gaussian functions with at least 300 variational parameters has been used as a trial function to guarantee an adequate convergence. In Ref. [13] the authors considered a structure formed by a 40 Å central well coupled through 20 Å thin barriers to lateral wells of 140 Å on both sides. The barriers heights for the lateral and central wells in the considered structure were taken as 200 and 50 meV, respectively. In Fig. 1(c) we present binding energies in such a structure calculated as functions of the impurity position for three different values of the intensity of magnetic field. The agreement between our results (solid lines) and those from Ref. [13] (symbols) is excellent. In Fig. 2 we plot the binding energies of three first s-like excited states as functions of the QW width for a donor placed at the center of the well. It is seen that the donor binding energies for large well widths descend slowly towards the corresponding well-known bulk exact values, i.e., to Ry∗ /4 for the Σ (2s) state, to Ry∗ /9 for the Σ (3s) state, and to Ry∗ /16 for the Σ (4s) state, respectively. On the opposite sides of the curves, as the well width decreases, the binding energies initially increase slightly but then drop abruptly tending to zero as the well width approaches 4a∗0 , i.e. when the distance from the donor to the barrier is equal to 2a∗0 . We associate this result with the impossibility of forming donor excited states, stable with respect to the ionization, when the first node in the z-direction for the wave functions is situated outside the
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a
293
b
. . . . . .
c
Fig. 1. Comparison of our calculation results for the ground state donor energies confined in different GaAs/Ga(Al)As heterostructures with those obtained previously by other methods: (a) the on-center donor binding energy as a function of a single QW width; (b) D0 energy as function of width of the barriers between the wells in a MQW system consisting of five 15 nm QWs for an impurity placed at the center of the central well; (c) D0 binding energies in a triple QW as a function of the impurity position for different values of the magnetic field. .
.
.
.
Fig. 2. On-center donor binding energies of three s-like excited states as a function of the GaAs/Ga0.3 Al0.7 As QW width. and (b) the applied electric field, for three donor positions.
well. It is worth mentioning that the hydrogen wave functions corresponding to the three excited states, whose curves are presented in Fig. 2, have the first node in the radial direction close the point r = 2a∗0 . In what follows we present results of the analysis of the effect of the electric field on the ground state of an on-axis donor in GaAs/Ga1−x Alx As SL and NWSL with a cylindrical cross-section of radius R and with variable concentration of Al along the crystal growth z direction, which is equal to zero inside the n identical wells of width a and to x in the n − 1 barriers of width b. We assume that the
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a
b
c
Fig. 3. Ground state energy of an on-axis donor in heterostructures with 11 unit cells (a = 10 nm, b = 5 nm) as a function of the impurity position, under different electric fields in (a) SL, and in NWSL of radii (b) R = 40 nm and (c) R = 10 nm.
confinement potential in such a structure in the z direction is given by a piecewise constant function V (z ), which is equal to zero in the wells and to V0 in the barriers. In Figs. 3–5 we present the results of our calculations for the GaAs/Ga0.7 Al0.3 As SL with a = 10 nm, b = 5 nm and for NWSL with the same parameters a and b and two different radii: 10, 40 nm. The effective Rydberg, Ry∗ , in these structures is about 5.83 meV, the Bohr radius, a∗0 , about 10 nm and the barrier height, V0 , about 40 Ry*. Dependencies of the donor energies on the distance ξ from the left end of the NWSL for several values of the external electric field are shown in Fig. 3. It is seen that for zero electric field the donor energy variation with the displacement of the donor position from the middle of the heterostructures towards one of the ends of the NWSL is wavy. This undulating behavior of the energy dependence is caused by the variation of the probability of finding the electron in different parts of the NWSL. The higher the probability to find the electron at the donor location, the stronger is the attraction between the electron and the ion, and the greater is the drop in energy. When electric field is absent, the greatest probability of finding the electron is at the middle of the central well and therefore at this point the energy is minimal. When the donor is moved from this point towards one of the NWSL ends the donor energy E D0 is increased due to the increasing average separation between the electron
and the ion, until E D0 reaches a local maximum at the center of the first adjacent barrier, where the probability to find the electron is smaller than in the well. As the donor is moved further toward the NWSL end, it enters the neighboring well, and the energy is decreased again until the donor reaches the center of this well. Since the density in the neighboring well is smaller than in the central well, the new local minimum is higher than main minimum in the central well. Also, one can observe in Fig. 3 that the energy dependence is altered essentially in the presence of the external electric field. The undulating character of the ground state energy dependence disappears and the position of the main energy minimum is displaced sharply to the center of the first well. Comparing Fig. 3(a)–(c) one can see that as the NWSL radius decreases a stronger electric field should be applied in order to remove the undulating energy dependencies.
W. Gutiérrez et al. / Superlattices and Microstructures 48 (2010) 288–297
a
295
b
. . . . . . .
c
. . . . . .
Fig. 4. Ground state energy of a donor located at the center of the central well in the GaAs/Ga0.7 Al0.3 As (a) SL, and NWSL of radii (b) R = 40 nm and (c) R = 10 nm with different numbers of wells as a function of the electric field.
There is a strong correlation between the probability density function of the electron in the heterostructure and the donor binding energy. Basically, a higher the probability of finding the electron at the donor location leads to the electron spending more time close to the donor, therefore resulting in a higher binding energy. The greater the radius of the NWSL or the number of wells, the weaker is the localization of the electron inside the wells and the easier it is for the electron to be displaced far from the donor location under an electric field. In order to reveal this fact we present in Fig. 4 the calculation results for the binding energies of donors located at the center of the central well as functions of the electric field strength in SL and two NWSLs of radii 40 and 10 nm and with one, three, five, and eleven wells. As the electric field increases, the electron is pulled away from the ion toward one of the ends of the heterostructure, the electron wave function becomes more extended, and the donor binding energy drops, due to an increase in the average distance between the electron and ion, until it reaches a minimum, when the electron is mainly located close to one of the ends of the heterostructure. One can see in Fig. 4 that both the slope of the initial part of curves and the minimum value of the binding energy depend on the number of wells and the radius of the NWSL. The greater the number of the wells, the longer the wire, the stronger the lowering of the potential energy due to the electric field, and the easier it is to pull the electron away from the ion, resulting in an increase of the slope in the curve of the binding energy dependence on the electric field. A similar change of the slope of the curves can be observed in Fig. 4 with an increase of the radius of the NWSL. An increase of the NWSL radius provides a reduction of the electron confinement inside the wells and leads to a stronger tunneling of the electron along the axis under an external electric field applied along the axis, resulting in an increase of the slope of the curves, as observed in Fig. 4(a)–(c). Dependencies of the binding energies on the electric field for an impurity located at the center of the first well of the NWSL are shown in Fig. 5. In this case the electron is pulled toward the donor under an external electric field and therefore the binding energy increases when the electric field is increased. In contrast to the dependencies presented in Fig. 4, the binding energies in Fig. 5 in a strong electric field tend to the same value for different numbers of the wells in the heterostructure. This is due to the fact that the electron under a strong electric field is pulled inside the same cell where the donor is located and therefore the average separation between the electron and ion does not depend on the number of wells.
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a
b .
. . .
.
. .
.
.
c
. . . . . .
Fig. 5. Ground state energy of a donor located at the center of the first well in the GaAs/Ga0.7 Al0.3 As (a) SL, and NWSL of radii (b) R = 40 nm and (c) R = 10 nm with different numbers of wells as a function of the electric field.
4. Summary and conclusions In this work we have proposed a new nomenclature for different electronic states of a donor confined in a semiconductor heterostructure with axial symmetry, and have presented a new simple method to calculate the corresponding wave functions and energies, based on the similarity of the ‘‘cusp’’ conditions for a donor confined in a heterostructure and those for a hydrogenic atom. The energy levels in our nomenclature are labeled as Λ(nl) , characterized by one ‘‘good’’ quantum number Λ, the projection of the orbital angular momentum on the symmetry axis, and two quantum numbers, n and l, which are considered as ‘‘good’’ only inside a restricted region close to the donor position, where the influence of the confinement is negligible. The donor eigenfunctions are represented as a product of the corresponding hydrogenic atom orbitals with an envelope function which takes into account the modification of these orbitals due to the confinement along and across the symmetry axis. Starting from the Schrödinger variational principle we derive an universal differential equation for this envelope function, applicable to any heterostructure with axial symmetry. The method has been used to calculate the ground and some excited energy states of donors confined in single and multiple quantum wells, and nanowire-superlattices for different magnetic and electric fields. The accuracy of our procedure is verified by the good accordance between our results for a single and multiple quantum wells with those previously obtained by using the Monte Carlo, finite difference and variational methods. One of the attractive features of our method is its universality and the simplicity of its application to different models of confinement in the framework of the same numerical procedure. It permits one to avoid tedious calculations in order to obtain, with a high accuracy, the donor energies in any heterostructure once the confinement potential is known. We present new results of application of the method to cylindrical GaAs–(Ga,Al)As nanowiresuperlattices in order to analyze the effect of the variation of the number of wells, the wire radius, the donor position, and the electric field strength. Similarly, problems for excitons or for other carrier complexes, confined in heterostructures with axial symmetry can be treated. Thus, our method can be applied to study the properties of different bound carrier states (neutral and negatively charged
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donors, acceptors, excitons, biexcitons, trions and so on) confined in different types of heterostructure in the presence of any external perturbation, and it may even exhibit practical advantages, similar to those shown in our work. Acknowledgements This work was financed by the Industrial University of Santander (UIS) through the Dirección General de Investigaciones (DIF Ciencias, Cod. 5111) and the Excellence Center of Novel Materials— ECNM, under Contract No. 043-2005 and the Cod. No. 1102-05-16923 subscribed with COLCIENCIAS. References [1] M. Grundmann (Ed.), Nano-Optoelectronics: Concepts, Physics and Devices, Springer-Verlag, Berlin, 2002. [2] G. Bastard, Phys. Rev. B24 (1981) 4714; V. Branis, G. Li, K.K. Bajaj, Phys. Rev. B 47 (1993) 1316; S.T. Pérez-Merchancano, G.E. Marques, Solid State Commun. 110 (1999) 209. [3] T. Pang, S.G. Louie, Phys. Rev. Lett. 65 (1990) 1635. [4] S. Glutsch, Excitons in Low-Dimensional Semiconductors, Springer Verlag, 2004. [5] J.L. Zhu, J.H. Zhao, J.J. Xiong, Phys. Rev. B 50 (1994) 1832; A. Gonzalez, I. Mikhailov, Int. J. Mod. Phys. B 11 (1997) 3469; M. El-Said, Physica B 202 (1994) 202. [6] R.L. Greene, K.K. Bajaj, Solid State Commun. 45 (1983) 825; Phys. Rev. B 31 (1985) 913; A. Latge, N. Porras-Montenegro, L.E. Oliveira, Phys. Rev B. 45 (1992) 9420. [7] S. Chaudhuri, K.K. Bajaj, Phys. Rev. B 29 (1984) 1803; H. Sari, E. Kasapoglu, I. Sokmen, Physica B 325 (2003) 300. [8] M. Helm, W. Hilber, T. Fromherz, F.M. Peeters, K. Alavi, R.N. Pathak, Phys. Rev. B 48 (1993) 1601; M. Helm, F.M. Peeters, F. DeRosa, E. Colas, J.P. Harbison, L.T. Florez, Phys. Rev. B 43 (1991) 13983; J.M. Shi, F.M. Peeters, G.Q. Hai, J.T. Devreese, Phys. Rev. B 44 (1991) 5692; J.M. Shi, F.M. Peeters, G.Q. Hai, J.T. Devreese, Phys. Rev. B 50 (1994) 15182. [9] F.J. Betancur, I.D. Mikhailov, L.E. Oliveira, J. Phys. D 31 (1998) 3391. [10] S. Fraizzoli, F. Bassani, R. Buczko, Phys. Rev. B 41 (1990) 5096. [11] I.D. Mikhailov, F.J. Betancur, R.A. Escorcia, J. Sierra-Ortega, Phys. Rev. B 67 (2003) 115317. [12] T. Kuhn, G. Mahler, J.L. Dunn, C.A. Bates, J. Phys.: Condens. Matter 6 (1994) 757. [13] M. Pacheco, Z. Barticevic, A. Latgé, Physica B 302 (2001) 77.