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Effective carrier interaction in semiconductor thin films: A model-independent formula
ARTICLE IN PRESS Physica E 42 (2010) 1633–1636
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Effective carrier interaction in semiconductor thin films: A model-independent formula Nenad S. Simonovic´ Institute of Physics, University of Belgrade, P.O. Box 57, 11001 Belgrade, Serbia
a r t i c l e in fo
abstract
Article history: Received 14 December 2009 Accepted 11 January 2010 Available online 15 January 2010
It is shown that the effective carrier interaction in semiconductor thin films, which is essentially of a non-Coulomb type, depends on the layer thickness but it is not sensitive to the form of quantum well. As a consequence the analytical expression for the effective 2D interaction potential, obtained using the parabolic quantum well model, can be used as a general (model-independent) formula. As an example, we have considered the electrons localized in a quantum dot. It is demonstrated that, when the quantum well confinement is much stronger than the lateral one, the results obtained using the 2D approach with the effective potential are in a good agreement with the full 3D calculations. & 2010 Elsevier B.V. All rights reserved.
Keywords: Semiconductor Thin film Quantum well Quantum dot Effective potential
1. Introduction Fast development of the semiconductor technology has made possible to fabricate the heterostructures consisting of various thin ð 10 nmÞ semiconductor/insulator layers (films). By inserting a semiconductor between two insulator layers, one obtains a realization of the quantum well for electrons (and holes) in the semiconductor (see e.g. Refs. [1,2]). The thickness of this quantum well is for an order of magnitude or more larger than the lattice constant, allowing the effective mass approximation, but it is small enough that at low temperatures only the quantum state with the lowest energy e1 (the ground state of the quantum well) is occupied by electrons (see the bottom of Fig. 1). Then, in the direction perpendicular to the film (z-axis) the electrons in the semiconductor perform only the zero-point motion, i.e. essentially they can only move laterally (xy-plane) in the layer. If we additionally confine this two-dimensional (2D) electron gas laterally, we shall obtain the ’zero-dimensional’ system called quantum dot (QD), see Fig. 1. For the electrons dynamics in a QD, beside the full 3D confinement, the electron-electron (e-e) correlations are essential (see e.g. Ref. [5,3,4] for a review). For this reason these systems are sometimes called ’artificial atoms’. Since the lateral size of QDs created in thin films is typically few hundreds nanometers, i.e. for an order of magnitude larger than the layer thickness, the concept of 2D electron gas can be extended to these systems. As a consequence the usual theoretical approach for QDs formed in thin film
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heterostructures is two-dimensional [6,7]. A similar conclusion holds for the electron-hole (e-h) bound systems (excitons) created in such layers. If the exciton Bohr radius is much larger than the thickness d (the so-called strong confinement regime) the 2D approximation may be satisfactory [1,2]. The 2D approach, of course, breaks down when the effective lateral confinement becomes comparable to the perpendicular one. Then the full three-dimensional (3D) approach becomes necessary [10,11]. In typical samples, however, the thickness effects can be included either using the full 3D approach (see e.g. Ref. [12] for two-electron QDs) or through the effective e-e (or e-h) interaction within the 2D model (the quasi-3D model, see below). A modification of the latest approximation when the interaction between electrons keeps the Coulomb-type form, the so-called effective charge approximation, has been considered recently in the case of two-electron QDs [13,10].
2. Quantum well models and the probability distribution for the zero point motion Certainly, the simplest quantum well models are: (i) the onedimensional (1D) infinite square well (the hard wall model) ( 0; jzjo d=2 ð1Þ V? ¼ 1; jzjZ d=2; where d is the layer thickness, and (ii) the parabolic well V? ¼ 12 m o2? z2 (linear harmonic oscillator), where m and o? are the electron (hole) effective mass and the perpendicular characteristic frequency, respectively. These two potentials can be understood as the hard/soft wall limiting cases keeping in mind
ARTICLE IN PRESS N.S. Simonovic´ / Physica E 42 (2010) 1633–1636
1634
lateral confinement
x,y
z
quantum well
ε1
d/2
V⊥
QD
− d/2 Fig. 1. The localization of a QD in the semiconductor layer of the thickness d (bottom left) and schematic plots showing the corresponding lateral (top) and perpendicular (right) confinements, as well as the lowest levels (thin orange/red lines) and the probability distributions (thick orange/red lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
V⊥ |ϕ|2
V⊥ |ϕ|2
εosc 1 ε1
ε1 −d/2 -z0
0
z 0 d/2
z
-d/2 -z0
0
z 0 d/2 z
Fig. 2. The ground state energy levels (e1 ) and the corresponding probability distributions (thick lines) for the square well (red lines) and parabolic (blue/green lines) models in the cases: (a) when o? ¼ o1 and (b) when o? is given by Eq. (3) with c ¼ 3:85. An example of the quantum well (black dashed line), which form is between the parabolic and the square well, and the corresponding ground state probability distribution (yellow dashed line) are shown for the case (a). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
that a more realistic model may have a form between (i) and (ii) (see Fig. 2a). The hard wall model is widely used because it is able to describe thickness effects in the majority of typical samples. Contrary, the parabolic one corresponds rather to specific samples based, for example, on GaAs=Alx Ga1x As heterostructures where the parabolic shape is caused by the varying Al content x along the growth direction (z-axis) [8,9]. From the mathematical point of view, however, the second model is more suitable because in many cases it provides analytical solutions (particularly for QDs where the lateral confinement is also parabolic). For this reason in this paper we inspect the applicability of the parabolic model for typical samples, too, and compare the results obtained in these two (i/ii) cases. Unfortunately, there is no general relation between the parameter o? and the layer thickness d. Namely, one fixed value for o? can be used only in a restricted energy domain. In the following we show how the value o? can be estimated from the layer thickness d, under the assumption that electrons occupy only the ground state of the well. The period of classical motion for a particle of the mass m and e confined in the square well defined by Eq. (1) is energy pffiffiffiffiffiffiffiffiffiffiffiffiffiffi the frequency of this periodic motion is T ¼ d 2m =e. Thus, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi o ¼ 2p=T ¼ ðp=dÞ 2e=m . Since the lowest energy level in the 1D 2 infinite square well is e1 ¼ p2 ‘ =ð2m d2 Þ, the associated frequency will be
o1 ¼
p2 ‘ m d2
ð2Þ
and we can write e1 ¼ 12 ‘ o1 . Clearly, if we choose in the parabolic model the parameter o? ¼ o1 , two models will be characterized by the same ground state energy. Besides, for this choice, the
probability distributions jjðzÞj2 , where jðzÞ are the corresponding ground state wave functions, are also similar (see Fig. 2 a). Emphasize that the corresponding distribution for any quantum well, which form is between the limiting cases (i) and (ii), will be also close to these two (i/ii) distributions. An example is the potential V? ¼ ðA=d2 Þtan2 ðkpz=dÞ (d=2k o z od=2k) which is for certain values of the parameters A and k (a continuous family) characterized by the same ground state energy e1 as the models (i) and (ii). (Note that for A-0 and k-1 this potential reduces to the infinite square well potential.) The latest potential, for A ¼ 2 and k ¼ 0:8435, and the corresponding probability distribution are shown in Fig. 2 a (dashed lines). Even better agreement between the probability distributions for the models (i) and (ii) can be obtained if we choose
o? ¼
c2 ‘ m d2
ð3Þ
with c 3:85 (see Fig. 2 b). For this value the ground state probability distribution approximately vanishes at z ¼ 7d=2 in the parabolic model, too. (In fact, the parameter c is matched precisely to this value after introducing the screening function below.) Note that relation (3) finally reduces to d ¼ cz0 , where ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z0 ¼ ‘ =m o? is the harmonic oscillator characteristic length. A disadvantage is that the ground state energy in the latest case overestimates the square well value, i.e. 12 ‘ o? 4 12 ‘ o1 (see Fig. 2 b). However, since for small d the electrons occupy only the lowest level of the quantum well (at least at low temperatures), this zero-point-motion energy will produce only a constant energy shift in the total energy.
3. Dynamical screening Another, less trivial, effect of the sample thickness is the reduction (dynamical screening) of the Coulomb interaction (e-e/e-h) comparing to that in the pure 2D model [13]. In the following we show that, if the condition (3) holds, the considered two models give almost identical screening rates. In order to justify this statement let us consider N electrons in a semiconductor layer, which are additionally confined laterally by a parabolic trap with the characteristic frequency o0 5 o? , i.e. consider an anisotropic axially symmetric QD (see Fig. 1). If we add, besides, a perpendicular magnetic field the lateral confinement remains the parabolic, but with the effective frequency O ¼ ðo20 þ o2L Þ1=2 which depends on magnetic field through the Larmor frequency oL ¼ eB=2m . Another effect of the field is P the constant term oL Lz in the Hamiltonian, where Lz ¼ i lzi is the z-projection of the total angular momentum and lzi are the projections of the individual electrons angular momenta. In principle the lateral and perpendicular motions may be coupled by the confining potential [14]. However, for the sake of simplicity we shall assume that this coupling can be neglected. Then the corresponding 3D Hamiltonian (using cylindrical coordinates) reads H¼
N X i¼1
p2ri
l2zi
1 þ þ m O2 r2i oL lzi 2 2m 2m r2i
! þ
N X i¼1
! p2zi þ V? ðzi Þ þ VC ; 2m
ð4Þ ðx2i þy2i Þ1=2 ,
where ri ¼ ji ¼ arctanðyi =xi Þ, zi are the i-th electron coordinates and pri , pji lzi =ri , pzi are the conjugated momenta. In that case the coupling between the lateral and perpendicular motions comes only from the Coulomb term describing the interaction (repulsion) between electrons VC ¼
N N X X k ; r i ¼ 1 j ¼ i þ 1 ij
ð5Þ
ARTICLE IN PRESS N.S. Simonovic´ / Physica E 42 (2010) 1633–1636
where k ¼ e2 =ð4pe0 er Þ (er is the relative dielectric constant of the semiconductor). Here rij ¼ ðr2ij þ z2ij Þ1=2 , rij ¼ ½ðxi xj Þ2 þ ðyi yj Þ2 1=2 and zij ¼ zi zj . If the first and the second sum in Eq. (4) we denote by H0 and H? , respectively, this Hamiltonian reads concisely H ¼ H0 þ H? þ VC . In real samples V? is much stronger than the lateral confinement. It results in different time scales of the lateral and perpendicular motions and allows one to use the adiabatic approach which effectively decouples them. This approach consists of averaging the full 3D Hamiltonian over the fast perpendicular motion. We have shown previously how it can be done efficiently by the use of the action-angle variables [13]. In the present work, however, different quantum well models are related through their ground state probability distributions and, for this reason, the effective 2D Hamiltonian (the quasi-3D model) is constructed by averaging the full 3D Hamiltonian over the P ground state of H? (jgr ¼ i jðzi Þ) Heff /HS? ¼ H0 þN e1 þ /VC S? ;
ð6Þ
where / S? /jgr j jjgr S. The effective 2D interaction P potential /VC S? ¼ i;j k/rij1 S? is a function of the rij variables, which is essentially different from the 2D Coulomb form P i;j r1 ij . (Within the 2D frame we can talk about a dynamical
characteristic length z0 is defined above) and the double integral in /rij1 S? can be expressed in terms of the modified Bessel function of the second kind K0 , giving ! ! rij 2 rij 2 1 rij f ðrij Þ ¼ pffiffiffiffiffiffi K0 : ð9Þ exp 2z0 2z0 2p z0 Since z0 ¼ d=c (see Eq. (3) and the text below) this screening function, like in the previous case, depends on the dimensionless argument rij =d. The function is shown in Fig. 3 for the values c ¼ p (full blue line) and c ¼ 3:85 (dashed green line), together with the function (8). It can be seen that the functions (8) and (9) when c ¼ p are very close, but they practically coincide if c ¼ 3:85.
4. Applications and conclusions A valuable consequence of the latest is that the analytical expression (9) can be used in the effective Hamiltonian (6) independently on the quantum well model. This is true not only for the limiting cases (i) and (ii) but also for other (possibly more realistic) models, which V? ðzÞ form is between these two cases, because the probability distributions jjðzÞj2 for these models and
screening of the Coulomb interaction by the perpendicular zerofor rij b d (see the point motion.) However, since /rij1 S? -r1 ij
24
inset in Fig. 3), it is convenient to write /VC S? ¼
X kf ðrij Þ
rij
i;j
;
1635
22
ð7Þ
where f ðrij Þ ¼ r
¼ rij /ji ; j
ji ; jj S
1 j jrij j
is the screening function. (i) For the 1D infinite pffiffiffiffiffiffiffiffi square well (1) the ground state wave function is jðzÞ ¼ 2=d cosðpz=dÞ and rij Z 1=2 Z 1=2 cos2 ðpzi Þcos2 ðpzj Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dzi dzj ; ð8Þ f ðrij Þ ¼ 4 d 1=2 1=2 ðr =dÞ2 þ ðz z Þ2 i j ij
Erel (meV)
20 1 ij /rij S?
m=3
18
m=2
16
m=1
14 12
where z z=d. This integral is evaluated numerically for different values of rij =d and the screening function is shown in Fig. 3 (dashed red line). (ii) For the parabolic potential, however, the ground state wave function has the form jðzÞ ¼ ðpz20 Þ1=4 expðz2 =2z20 Þ (the
10
1
m=0 0
2
4
6 B (T)
8
10
12
8
0.8 0.6
Δμ (meV)
6/d 1/ρ
f
f/ρ
4/d
0.4
2/d
m=2
7.5 m=1 7 m=0
0.2 0
0
0
0
d
d
ρ
ρ
2d
2d
6.5
3d
3d
Fig. 3. The screening function f ðrÞ r /r1 S? calculated using: (i) the hard-wall model (dashed red line) and (ii) the parabolic model for c ¼ p (full blue line) and for c ¼ 3:85 (dashed green line). The inset shows the effective potential f ðrÞ=r for the same cases. The pure Coulomb (2D) potential (black dotted line) is shown for the comparison. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
0
2
4
6 B (T)
8
10
12
Fig. 4. (a) The lowest energy levels for the electrons relative motion with m ¼ 0; 1; 2 and 3 of the GaAs (m ¼ 0:067me , er ¼ 12) two-electron QD characterized by ‘ o0 ¼ 5 meV and ‘ o? ¼ 40 meV (related by Eq. (3) to the sample thickness d 20:5 nm) as functions of the strength of perpendicular magnetic field B. The results obtained in the 2D, the full 3D and the quasi-3D approach are drawn using dotted, dashed, and full (orange) lines, respectively. (b) The additional energy for the second electron in the dot, DmðBÞ ¼ mð2; BÞmð1; BÞ, calculated within the same three approaches. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
ARTICLE IN PRESS 1636
N.S. Simonovic´ / Physica E 42 (2010) 1633–1636
the models (i) and (ii) are almost identical. Consequently, the energy spectra obtained by solving the eigenvalue problem ðHeff EÞC ¼ 0 in the cases (i) and (ii) will distinguish only by the energy shift N e1 . If we keep the number of electrons in the dot fixed, this shift has no important role and may be dropped. In other words, the structure of the energy spectrum of the considered N-electron system depends on the layer thickness but it is not sensitive to the form of quantum well. Remark that this approach (with the same screening function, Eq. (9)) can be used also to study the influence of the well width to the exciton spectrum within the strong confinement regime. To demonstrate the relevance of the proposed quasi-3D model (i.e. the 2D model with the effective e-e (or e-h) interaction) we show in Fig. 4a the energy spectra (lowest levels for the relative motion) of the two-electron QD in the GaAs (m ¼ 0:067 me , er ¼ 12) layer obtained using: the 2D (dotted line), the full 3D (dashed line) and the quasi-3D (full orange line) approaches. The levels are labelled by the quantum number m of the z-projection of angular momentum for the relative motion. In this example we choose for the lateral confinement ‘ o0 ¼ 5 meV in all three cases. Particularly, in the 3D calculation we have used the fully parabolic confining potential (i.e. parabolic lateral confinement + parabolic quantum well) with this ‘ o0 value and ‘ o? ¼ 40 meV. The same value for o? , which is related by Eq. (3) to the layer thickness d 20:5 nm, is used to calculate the effective interaction potential (i.e. the screening function (9)) for the quasi-3D model. For a better comparison with the pure 2D results the energy shift e1 , which appears in the latest two approaches, is dropped. One can observe a remarkable agreement (especially for m a0) between the results obtained in the full 3D and quasi-3D approaches. The most direct probe of electron correlation in a QD is the difference of the electrochemical potentials DmðNÞ ¼ mðNÞ mðN1Þ. The electrochemical potential inside the dot with N electrons equals mðNÞ ¼ Egr ðNÞEgr ðN1Þ, where Egr ðNÞ is the ground-state energy of the dot with N electrons. Therefore, mðNÞ is the energy necessary to add the N th electron into the dot and it can be determined experimentally, for example, by measuring the gate voltage when these two quantities are in resonance (see e.g. Refs. [3–5]). Since the electrochemical potential increases with N because of the increasing Coulomb repulsion, the difference DmðNÞ is the additional energy required to overcome this increment. It is not difficult to check that DmðNÞ does not depend on e1 because
the corresponding energy shifts cancel. In Fig. 4 b we show the additional energy Dmð2Þ for the same two-electron QD which energy spectrum is shown in Fig. 4 a, calculated using the 2D, the full 3D and the quasi-3D approaches. It can be seen that the inclusion of the finite layer thickness changes noticeable the 2D result. As in the case of the energy spectrum, the results for the additional energy obtained in the full 3D and quasi-3D approaches are in a good agreement. Summarizing, we have shown that two simple quantum well models, the hard wall and the parabolic one, as well as the models in between, which we have used to study the thickness effects in thin semiconductor films, give almost identical corrections to the 2D description of the carrier dynamics (the quasi-3D approach). As a consequence the analytical expression (9) for the screening function can be used independently on the quantum well model. It is demonstrated that, when the quantum well confinement is much stronger than the lateral one, the results obtained using the quasi-3D approach are in a good agreement with the full 3D calculations.
Acknowledgment This work was supported by Project no. 141029 of Ministry of Science and Technological Development of Serbia.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
S. Schmitt-Rink, D.S. Chemla, D.A.B. Miller, Adv. Phys. 38 (1989) 89. R. Cingolani, K. Ploog, Adv. Phys. 40 (1991) 535. L.P. Kouwenhoven, D.G. Austing, S. Tarucha, Rep. Prog. Phys. 64 (2001) 701. S.M. Reimann, M. Manninen, Rev. Mod. Phys. 74 (2002) 1283. R.C. Ashoori, Nature 379 (1996) 413. T. Chakraborty, Quantum Dots, Elsevier, Amsterdam, 1999. P.A. Maksym, H. Imamura, G.P. Mallon, H. Aoki, J. Phys.: Condens. Matter 12 (2000) R299. E.G. Gwinn, R.M. Westervelt, P.F. Hopkins, A.J. Rimberg, M. Sundaram, A.C. Gossard, Phys. Rev. B 39 (1989) 6260. T. Ihn, C. Ellenberger, K. Ensslin, C. Yannouleas, U. Landman, D.C. Driscoll, A.C. Gossard, Int. J. Mod. Phys. B 21 (2007) 1316. R.G. Nazmitdinov, N.S. Simonovic´, J.-M. Rost, Phys. Rev. B 65 (2002) 155307. N.S. Simonovic´, R.G. Nazmitdinov, Phys. Rev. B 67 (2003) 041305 (R). R.G. Nazmitdinov, N.S. Simonovic´, Phys. Rev. B 76 (2007) 193306. N.S. Simonovic´, R.G. Nazmitdinov, Phys. Rev. A 78 (2008) 032155. ¨ P. Junker, U. Kops, U. Merkt, T. Darnhofer, U. Rossler, Phys. Rev. B 49 (1994) 4794.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Effective carrier interaction in semiconductor thin films: A model-independent formula Nenad S. Simonovic´ Institute of Physics, University of Belgrade, P.O. Box 57, 11001 Belgrade, Serbia
a r t i c l e in fo
abstract
Article history: Received 14 December 2009 Accepted 11 January 2010 Available online 15 January 2010
It is shown that the effective carrier interaction in semiconductor thin films, which is essentially of a non-Coulomb type, depends on the layer thickness but it is not sensitive to the form of quantum well. As a consequence the analytical expression for the effective 2D interaction potential, obtained using the parabolic quantum well model, can be used as a general (model-independent) formula. As an example, we have considered the electrons localized in a quantum dot. It is demonstrated that, when the quantum well confinement is much stronger than the lateral one, the results obtained using the 2D approach with the effective potential are in a good agreement with the full 3D calculations. & 2010 Elsevier B.V. All rights reserved.
Keywords: Semiconductor Thin film Quantum well Quantum dot Effective potential
1. Introduction Fast development of the semiconductor technology has made possible to fabricate the heterostructures consisting of various thin ð 10 nmÞ semiconductor/insulator layers (films). By inserting a semiconductor between two insulator layers, one obtains a realization of the quantum well for electrons (and holes) in the semiconductor (see e.g. Refs. [1,2]). The thickness of this quantum well is for an order of magnitude or more larger than the lattice constant, allowing the effective mass approximation, but it is small enough that at low temperatures only the quantum state with the lowest energy e1 (the ground state of the quantum well) is occupied by electrons (see the bottom of Fig. 1). Then, in the direction perpendicular to the film (z-axis) the electrons in the semiconductor perform only the zero-point motion, i.e. essentially they can only move laterally (xy-plane) in the layer. If we additionally confine this two-dimensional (2D) electron gas laterally, we shall obtain the ’zero-dimensional’ system called quantum dot (QD), see Fig. 1. For the electrons dynamics in a QD, beside the full 3D confinement, the electron-electron (e-e) correlations are essential (see e.g. Ref. [5,3,4] for a review). For this reason these systems are sometimes called ’artificial atoms’. Since the lateral size of QDs created in thin films is typically few hundreds nanometers, i.e. for an order of magnitude larger than the layer thickness, the concept of 2D electron gas can be extended to these systems. As a consequence the usual theoretical approach for QDs formed in thin film
Tel.: + 381 11 3713 114; fax: + 381 11 3162 190.
E-mail address: [email protected] 1386-9477/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2010.01.012
heterostructures is two-dimensional [6,7]. A similar conclusion holds for the electron-hole (e-h) bound systems (excitons) created in such layers. If the exciton Bohr radius is much larger than the thickness d (the so-called strong confinement regime) the 2D approximation may be satisfactory [1,2]. The 2D approach, of course, breaks down when the effective lateral confinement becomes comparable to the perpendicular one. Then the full three-dimensional (3D) approach becomes necessary [10,11]. In typical samples, however, the thickness effects can be included either using the full 3D approach (see e.g. Ref. [12] for two-electron QDs) or through the effective e-e (or e-h) interaction within the 2D model (the quasi-3D model, see below). A modification of the latest approximation when the interaction between electrons keeps the Coulomb-type form, the so-called effective charge approximation, has been considered recently in the case of two-electron QDs [13,10].
2. Quantum well models and the probability distribution for the zero point motion Certainly, the simplest quantum well models are: (i) the onedimensional (1D) infinite square well (the hard wall model) ( 0; jzjo d=2 ð1Þ V? ¼ 1; jzjZ d=2; where d is the layer thickness, and (ii) the parabolic well V? ¼ 12 m o2? z2 (linear harmonic oscillator), where m and o? are the electron (hole) effective mass and the perpendicular characteristic frequency, respectively. These two potentials can be understood as the hard/soft wall limiting cases keeping in mind
ARTICLE IN PRESS N.S. Simonovic´ / Physica E 42 (2010) 1633–1636
1634
lateral confinement
x,y
z
quantum well
ε1
d/2
V⊥
QD
− d/2 Fig. 1. The localization of a QD in the semiconductor layer of the thickness d (bottom left) and schematic plots showing the corresponding lateral (top) and perpendicular (right) confinements, as well as the lowest levels (thin orange/red lines) and the probability distributions (thick orange/red lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
V⊥ |ϕ|2
V⊥ |ϕ|2
εosc 1 ε1
ε1 −d/2 -z0
0
z 0 d/2
z
-d/2 -z0
0
z 0 d/2 z
Fig. 2. The ground state energy levels (e1 ) and the corresponding probability distributions (thick lines) for the square well (red lines) and parabolic (blue/green lines) models in the cases: (a) when o? ¼ o1 and (b) when o? is given by Eq. (3) with c ¼ 3:85. An example of the quantum well (black dashed line), which form is between the parabolic and the square well, and the corresponding ground state probability distribution (yellow dashed line) are shown for the case (a). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
that a more realistic model may have a form between (i) and (ii) (see Fig. 2a). The hard wall model is widely used because it is able to describe thickness effects in the majority of typical samples. Contrary, the parabolic one corresponds rather to specific samples based, for example, on GaAs=Alx Ga1x As heterostructures where the parabolic shape is caused by the varying Al content x along the growth direction (z-axis) [8,9]. From the mathematical point of view, however, the second model is more suitable because in many cases it provides analytical solutions (particularly for QDs where the lateral confinement is also parabolic). For this reason in this paper we inspect the applicability of the parabolic model for typical samples, too, and compare the results obtained in these two (i/ii) cases. Unfortunately, there is no general relation between the parameter o? and the layer thickness d. Namely, one fixed value for o? can be used only in a restricted energy domain. In the following we show how the value o? can be estimated from the layer thickness d, under the assumption that electrons occupy only the ground state of the well. The period of classical motion for a particle of the mass m and e confined in the square well defined by Eq. (1) is energy pffiffiffiffiffiffiffiffiffiffiffiffiffiffi the frequency of this periodic motion is T ¼ d 2m =e. Thus, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi o ¼ 2p=T ¼ ðp=dÞ 2e=m . Since the lowest energy level in the 1D 2 infinite square well is e1 ¼ p2 ‘ =ð2m d2 Þ, the associated frequency will be
o1 ¼
p2 ‘ m d2
ð2Þ
and we can write e1 ¼ 12 ‘ o1 . Clearly, if we choose in the parabolic model the parameter o? ¼ o1 , two models will be characterized by the same ground state energy. Besides, for this choice, the
probability distributions jjðzÞj2 , where jðzÞ are the corresponding ground state wave functions, are also similar (see Fig. 2 a). Emphasize that the corresponding distribution for any quantum well, which form is between the limiting cases (i) and (ii), will be also close to these two (i/ii) distributions. An example is the potential V? ¼ ðA=d2 Þtan2 ðkpz=dÞ (d=2k o z od=2k) which is for certain values of the parameters A and k (a continuous family) characterized by the same ground state energy e1 as the models (i) and (ii). (Note that for A-0 and k-1 this potential reduces to the infinite square well potential.) The latest potential, for A ¼ 2 and k ¼ 0:8435, and the corresponding probability distribution are shown in Fig. 2 a (dashed lines). Even better agreement between the probability distributions for the models (i) and (ii) can be obtained if we choose
o? ¼
c2 ‘ m d2
ð3Þ
with c 3:85 (see Fig. 2 b). For this value the ground state probability distribution approximately vanishes at z ¼ 7d=2 in the parabolic model, too. (In fact, the parameter c is matched precisely to this value after introducing the screening function below.) Note that relation (3) finally reduces to d ¼ cz0 , where ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z0 ¼ ‘ =m o? is the harmonic oscillator characteristic length. A disadvantage is that the ground state energy in the latest case overestimates the square well value, i.e. 12 ‘ o? 4 12 ‘ o1 (see Fig. 2 b). However, since for small d the electrons occupy only the lowest level of the quantum well (at least at low temperatures), this zero-point-motion energy will produce only a constant energy shift in the total energy.
3. Dynamical screening Another, less trivial, effect of the sample thickness is the reduction (dynamical screening) of the Coulomb interaction (e-e/e-h) comparing to that in the pure 2D model [13]. In the following we show that, if the condition (3) holds, the considered two models give almost identical screening rates. In order to justify this statement let us consider N electrons in a semiconductor layer, which are additionally confined laterally by a parabolic trap with the characteristic frequency o0 5 o? , i.e. consider an anisotropic axially symmetric QD (see Fig. 1). If we add, besides, a perpendicular magnetic field the lateral confinement remains the parabolic, but with the effective frequency O ¼ ðo20 þ o2L Þ1=2 which depends on magnetic field through the Larmor frequency oL ¼ eB=2m . Another effect of the field is P the constant term oL Lz in the Hamiltonian, where Lz ¼ i lzi is the z-projection of the total angular momentum and lzi are the projections of the individual electrons angular momenta. In principle the lateral and perpendicular motions may be coupled by the confining potential [14]. However, for the sake of simplicity we shall assume that this coupling can be neglected. Then the corresponding 3D Hamiltonian (using cylindrical coordinates) reads H¼
N X i¼1
p2ri
l2zi
1 þ þ m O2 r2i oL lzi 2 2m 2m r2i
! þ
N X i¼1
! p2zi þ V? ðzi Þ þ VC ; 2m
ð4Þ ðx2i þy2i Þ1=2 ,
where ri ¼ ji ¼ arctanðyi =xi Þ, zi are the i-th electron coordinates and pri , pji lzi =ri , pzi are the conjugated momenta. In that case the coupling between the lateral and perpendicular motions comes only from the Coulomb term describing the interaction (repulsion) between electrons VC ¼
N N X X k ; r i ¼ 1 j ¼ i þ 1 ij
ð5Þ
ARTICLE IN PRESS N.S. Simonovic´ / Physica E 42 (2010) 1633–1636
where k ¼ e2 =ð4pe0 er Þ (er is the relative dielectric constant of the semiconductor). Here rij ¼ ðr2ij þ z2ij Þ1=2 , rij ¼ ½ðxi xj Þ2 þ ðyi yj Þ2 1=2 and zij ¼ zi zj . If the first and the second sum in Eq. (4) we denote by H0 and H? , respectively, this Hamiltonian reads concisely H ¼ H0 þ H? þ VC . In real samples V? is much stronger than the lateral confinement. It results in different time scales of the lateral and perpendicular motions and allows one to use the adiabatic approach which effectively decouples them. This approach consists of averaging the full 3D Hamiltonian over the fast perpendicular motion. We have shown previously how it can be done efficiently by the use of the action-angle variables [13]. In the present work, however, different quantum well models are related through their ground state probability distributions and, for this reason, the effective 2D Hamiltonian (the quasi-3D model) is constructed by averaging the full 3D Hamiltonian over the P ground state of H? (jgr ¼ i jðzi Þ) Heff /HS? ¼ H0 þN e1 þ /VC S? ;
ð6Þ
where / S? /jgr j jjgr S. The effective 2D interaction P potential /VC S? ¼ i;j k/rij1 S? is a function of the rij variables, which is essentially different from the 2D Coulomb form P i;j r1 ij . (Within the 2D frame we can talk about a dynamical
characteristic length z0 is defined above) and the double integral in /rij1 S? can be expressed in terms of the modified Bessel function of the second kind K0 , giving ! ! rij 2 rij 2 1 rij f ðrij Þ ¼ pffiffiffiffiffiffi K0 : ð9Þ exp 2z0 2z0 2p z0 Since z0 ¼ d=c (see Eq. (3) and the text below) this screening function, like in the previous case, depends on the dimensionless argument rij =d. The function is shown in Fig. 3 for the values c ¼ p (full blue line) and c ¼ 3:85 (dashed green line), together with the function (8). It can be seen that the functions (8) and (9) when c ¼ p are very close, but they practically coincide if c ¼ 3:85.
4. Applications and conclusions A valuable consequence of the latest is that the analytical expression (9) can be used in the effective Hamiltonian (6) independently on the quantum well model. This is true not only for the limiting cases (i) and (ii) but also for other (possibly more realistic) models, which V? ðzÞ form is between these two cases, because the probability distributions jjðzÞj2 for these models and
screening of the Coulomb interaction by the perpendicular zerofor rij b d (see the point motion.) However, since /rij1 S? -r1 ij
24
inset in Fig. 3), it is convenient to write /VC S? ¼
X kf ðrij Þ
rij
i;j
;
1635
22
ð7Þ
where f ðrij Þ ¼ r
¼ rij /ji ; j
ji ; jj S
1 j jrij j
is the screening function. (i) For the 1D infinite pffiffiffiffiffiffiffiffi square well (1) the ground state wave function is jðzÞ ¼ 2=d cosðpz=dÞ and rij Z 1=2 Z 1=2 cos2 ðpzi Þcos2 ðpzj Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dzi dzj ; ð8Þ f ðrij Þ ¼ 4 d 1=2 1=2 ðr =dÞ2 þ ðz z Þ2 i j ij
Erel (meV)
20 1 ij /rij S?
m=3
18
m=2
16
m=1
14 12
where z z=d. This integral is evaluated numerically for different values of rij =d and the screening function is shown in Fig. 3 (dashed red line). (ii) For the parabolic potential, however, the ground state wave function has the form jðzÞ ¼ ðpz20 Þ1=4 expðz2 =2z20 Þ (the
10
1
m=0 0
2
4
6 B (T)
8
10
12
8
0.8 0.6
Δμ (meV)
6/d 1/ρ
f
f/ρ
4/d
0.4
2/d
m=2
7.5 m=1 7 m=0
0.2 0
0
0
0
d
d
ρ
ρ
2d
2d
6.5
3d
3d
Fig. 3. The screening function f ðrÞ r /r1 S? calculated using: (i) the hard-wall model (dashed red line) and (ii) the parabolic model for c ¼ p (full blue line) and for c ¼ 3:85 (dashed green line). The inset shows the effective potential f ðrÞ=r for the same cases. The pure Coulomb (2D) potential (black dotted line) is shown for the comparison. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
0
2
4
6 B (T)
8
10
12
Fig. 4. (a) The lowest energy levels for the electrons relative motion with m ¼ 0; 1; 2 and 3 of the GaAs (m ¼ 0:067me , er ¼ 12) two-electron QD characterized by ‘ o0 ¼ 5 meV and ‘ o? ¼ 40 meV (related by Eq. (3) to the sample thickness d 20:5 nm) as functions of the strength of perpendicular magnetic field B. The results obtained in the 2D, the full 3D and the quasi-3D approach are drawn using dotted, dashed, and full (orange) lines, respectively. (b) The additional energy for the second electron in the dot, DmðBÞ ¼ mð2; BÞmð1; BÞ, calculated within the same three approaches. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
ARTICLE IN PRESS 1636
N.S. Simonovic´ / Physica E 42 (2010) 1633–1636
the models (i) and (ii) are almost identical. Consequently, the energy spectra obtained by solving the eigenvalue problem ðHeff EÞC ¼ 0 in the cases (i) and (ii) will distinguish only by the energy shift N e1 . If we keep the number of electrons in the dot fixed, this shift has no important role and may be dropped. In other words, the structure of the energy spectrum of the considered N-electron system depends on the layer thickness but it is not sensitive to the form of quantum well. Remark that this approach (with the same screening function, Eq. (9)) can be used also to study the influence of the well width to the exciton spectrum within the strong confinement regime. To demonstrate the relevance of the proposed quasi-3D model (i.e. the 2D model with the effective e-e (or e-h) interaction) we show in Fig. 4a the energy spectra (lowest levels for the relative motion) of the two-electron QD in the GaAs (m ¼ 0:067 me , er ¼ 12) layer obtained using: the 2D (dotted line), the full 3D (dashed line) and the quasi-3D (full orange line) approaches. The levels are labelled by the quantum number m of the z-projection of angular momentum for the relative motion. In this example we choose for the lateral confinement ‘ o0 ¼ 5 meV in all three cases. Particularly, in the 3D calculation we have used the fully parabolic confining potential (i.e. parabolic lateral confinement + parabolic quantum well) with this ‘ o0 value and ‘ o? ¼ 40 meV. The same value for o? , which is related by Eq. (3) to the layer thickness d 20:5 nm, is used to calculate the effective interaction potential (i.e. the screening function (9)) for the quasi-3D model. For a better comparison with the pure 2D results the energy shift e1 , which appears in the latest two approaches, is dropped. One can observe a remarkable agreement (especially for m a0) between the results obtained in the full 3D and quasi-3D approaches. The most direct probe of electron correlation in a QD is the difference of the electrochemical potentials DmðNÞ ¼ mðNÞ mðN1Þ. The electrochemical potential inside the dot with N electrons equals mðNÞ ¼ Egr ðNÞEgr ðN1Þ, where Egr ðNÞ is the ground-state energy of the dot with N electrons. Therefore, mðNÞ is the energy necessary to add the N th electron into the dot and it can be determined experimentally, for example, by measuring the gate voltage when these two quantities are in resonance (see e.g. Refs. [3–5]). Since the electrochemical potential increases with N because of the increasing Coulomb repulsion, the difference DmðNÞ is the additional energy required to overcome this increment. It is not difficult to check that DmðNÞ does not depend on e1 because
the corresponding energy shifts cancel. In Fig. 4 b we show the additional energy Dmð2Þ for the same two-electron QD which energy spectrum is shown in Fig. 4 a, calculated using the 2D, the full 3D and the quasi-3D approaches. It can be seen that the inclusion of the finite layer thickness changes noticeable the 2D result. As in the case of the energy spectrum, the results for the additional energy obtained in the full 3D and quasi-3D approaches are in a good agreement. Summarizing, we have shown that two simple quantum well models, the hard wall and the parabolic one, as well as the models in between, which we have used to study the thickness effects in thin semiconductor films, give almost identical corrections to the 2D description of the carrier dynamics (the quasi-3D approach). As a consequence the analytical expression (9) for the screening function can be used independently on the quantum well model. It is demonstrated that, when the quantum well confinement is much stronger than the lateral one, the results obtained using the quasi-3D approach are in a good agreement with the full 3D calculations.
Acknowledgment This work was supported by Project no. 141029 of Ministry of Science and Technological Development of Serbia.
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