InP quantum wires with different shapes

Accepted Manuscript Title: Numerical modelling of auto-organized InAs/ InP quantum wires with different shapes Authors: F. Zaouali, A. Bouazra, M. Said PII: DOI: Reference:

S0030-4026(17)31658-3 https://doi.org/10.1016/j.ijleo.2017.12.030 IJLEO 60152

To appear in: Received date: Accepted date:

27-10-2017 12-12-2017

Please cite this article as: Zaouali F, Bouazra A, Said M, Numerical modelling of autoorganized InAs/ InP quantum wires with different shapes, Optik - International Journal for Light and Electron Optics (2010), https://doi.org/10.1016/j.ijleo.2017.12.030 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Numerical modelling of auto-organized InAs/ InP quantum wires with different shapes

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F.Zaouali*1, A. Bouazra, M. Said Laboratoire de la Matière Condensée et des Nanosciences (LMCN), Département de Physique, Faculté des Sciences de Monastir, 5019 Monastir, Tunisia Corresponding author: F.Zaouali

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*1

E-mail address: [email protected] Tel: +216 99864205 Fax: +216 73484405

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Laboratoire de la Matière Condensée et des Nanosciences (LMCN)

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Département de Physique, Faculté des Sciences de Monastir, 5019 Monastir, Tunisia

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Abstract

In this paper, energy levels of electrons and holes for different shapes of isolated and

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self-organized InAs/InP quantum wire are calculated. We have used a theoretical method based on the coordinate transformation employing a newly analytical function to resolve the

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Schrödinger equation. The purpose of this work is to show how the shape of the quantum wire depend on the parameters introduced in the function and to discuss the effect of the wire

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height on the energy levels. The optical transitions energies are also investigated.

Keywords: InAs/InP, Coordinate transformation, FDM, Transition energy. 1

1. Introduction In the past few years, the low-dimensional semiconductor structures like quantum dots, quantum wires, and quantum wells have aroused much attention due to their fundamental properties and their wide range application such as light emitting diodes[1-3], solar cells [4,5], medicine [6,7]and quantum computing[8]. Among them, the quantum wires, the charge carriers (electrons and holes) are confined in two dimensions. The quantum confinement of

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carriers in QWs leads to spectrally sharp energy levels depending on the shape and the size of quantum wires. Using the growth techniques of semiconductor nanostructures like molecularbeam epitaxy [9], it was possible to obtain a certain type of cross-sections such as dome, lens-

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shaped, T-shaped [10], elliptical [11] and V-groove quantum wires [12-15]. These quantum wires have been obtained in a variety of materials such as GaAs/AlGaAs [16-19], InAs/InP

[20] and InGaAs/GaAs [21]. Many theoretical and numerical methods are used for modelling

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and simulating such different shapes and forms of isolated and self-organized quantum wires [22].

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In this paper, focus will be on the modelling of different shapes of InAs quantum wire

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embedded in InP matrix. We will also study its electronic properties, suggesting a numerical

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method based on the combination of coordinate transformation and the FDM. . In the second section, we will present the description of this method, exposing the

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choice of coordinate transformation which can model the structures of dome and lens-shaped quantum wire.

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In the third section, we will investigate the influence of the quantum wire shape and height on the electron and hole energy levels, and transition energies. In section 4, we will

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finalise by drawing the conclusions of this paper. 2. Theoretical method

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2.1 Theoretical Formulation In the following, we present a general and theoretical method for modelling a

semiconductor quantum wire with different geometrical shapes in order to investigate their properties. We start from the envelope function approximation (EFA) 2D Schrödinger equation for electrons (holes) in conduction (valence) band:

2  2

  1   *   x, y   V x, y  x, y   E x, y   m  x, y   2

(1)

We can obtain the electron (hole) energy levels E and their corresponding wave function

 ( x, y) by solving this equation. where  x, y  is the varying part of the total wave function, m* is the electron or hole effective mass , E is the confinement energy and V x, y  is the total potential which is given by:

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 2 k z2 V x, y    V p x, y  2 m*

V x, y  is the 2D potential profile determined by the conduction or valence band offset, we p

can write Eq. (1) as: 2 2

   1  ( x, y )  1  ( x, y )    2 k z2     V p ( x, y ) ( x, y )  E ( x, y ) * *  x m* x, y  x y m x, y y    2m 

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

(2)

The first step of the method consists of a selection of the coordinate transformation that

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can provide a good fit of the shape and size of the computational domain. Once the coordinate

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transformation x  xu, v  and y  yu, v  is chosen. Consequently, Eq. (2) becomes:

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   u x  v x     u  v x     u y  v y     u  v y     v y  y*   *  *  *    v x  x*   u y  * u x  *  v  m u m v  u  m u m v  v  m u m * v   u  m u m v   U  E 2 2

(3)

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

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where

  u  v      u x  vx x u x v x u v

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(4)

  u  v      u y  vy y u y v y u v

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(5)

In Eq. (3),  x(u, v), y(u, v)   (u, v) , U p x(u, v), y(u, v)  U p (u, v) while ux, uy, vx and vy are

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the elements of the Jacobian matrix Jx,y representing partial derivatives of the inverse functions u  ux, y  and v  vx, y  with respect to x and y. The Jacobian matrix Juv corresponding to the transformation x  xu, v  , y  yu, v  is an explicit function on u and v:

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x J uv   u  yu

xv  y v 

(6)

Here xu / yu and xv / y v denote the partial derivatives to u and v respectively. Jxy is the Jacobian matrix for the inverse mapping u = u(x,y) and v = v(x,y), which is an explicit function on u and v. 1 It is given by J xy  J uv , we finally derive Jxy as:

u y   xu  v y   yu

xv  yv 

1

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J xy

u x  v x

(7)

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Therefore, the coordinate transformation which can fit lens and dome shapes and determine the computational domain can be written in this general form:

x  Ku y  Kf u, v 

(8a) (8b)

xv   K  y v   Kf u

0  Kf v 

u x J xy   v x

u y   1/ K 0  1 1    v y   f u /( Kf v ) 1 /( Kf v ) K 

(9a)

0  

(9b)

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x J uv   u  yu

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We note that K is a factor related to the scale and expressed in nanometers. Thus, we obtain:

Where fu and fv are derivatives of f(u,v) with respect to u and v.

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  K .v x   f u / f v and   K .v y  1 / f v

By using the expressions of  and  , the Eq. (3) can be expressed as follows :

 2    1       1                       U  E  v  m * u m * v  v  m * v  2 K 2  u  m * u m * v 

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

(10)

In the second step, the above equation is solved using 2D finite difference method and the Eq.

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(10) becomes:

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 

2 2K 2

1   hu

 1   i 1, j   i , j  1   i , j   i 1, j     i 1 / 2, j 1 / 2   i 1 / 2, j 1 / 2     i 1 / 2, j 1 / 2   i 1 / 2, j 1 / 2   *  *  *  *   hu hu hv hv  m i 1 / 2, j  m i 1 / 2, j  m i 1 / 2  m i 1 / 2, j 

 i , j 

 i 1 / 2, j 1 / 2   i 1 / 2, j 1 / 2  1   i 1 / 2, j 1 / 2   i 1 / 2, j 1 / 2     i , j 1   i , j     i , j   i , j 1  1   *  *  *  *   hu hu hv hv  m i , j 1 / 2  m i , j 1 / 2  m i , j 1 / 2  m i , j 1 / 2 

hv 

 i , j 

 

  hv  m* i , j 1 / 2 

 i , j 1   i , j hv

  i , j   i , j 1       *    U i , j i , j  E i , j hv  m i , j 1 / 2  

(11)

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Where i and j are indices of the discretization points, while hu is the discretization step along u coordinate and hv along v coordinate.

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2.2 coordinate transformation

In this section, we propose to study the coordinate transformation which is given as follows:

y  Kf u, v  

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x  Ku

(1  Bv 2 )

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))

(12)

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D

arctan h( K sin (

u

A

2

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sinh( Av )

Fig. 1 shows the manner each parameters in coordinate transformation (12) influences

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the shape of the function f u, v  , for a fixed value of v  2 . In Fig. 1a).b), we can see that A affects the peak of the function, which increases with A but decreases when the D parameter augments. The B parameter controls the shape width of quantum wire. An increase in B leads

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to a decrease in the curve width as shown in Fig. 1c). As can be seen in Fig. 1d), for fixed

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values of A, D and B, the maximum of the curve increases with v . The function given by relation (12) has the advantage of modelling one or more

 u  quantum wires, the part sin 2   is related to the periodicity, which gives us more than one  L

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quantum wire. We have taken, for example, three and five quantum wires and we suggest two models of this structure. Where    x, y  represents the old coordinate space, the Schrödinger equation can be mapped into new    u, v  space. In Figures 2a) and 2b), we have presented the two models of the quantum wire. The first model, shown in Fig. 2a), has a dome-like shape. In addition, the second model in Fig. 5

2a) has the form of lens cross-section. We observe from the figures that the two models are obtained when the parameter v varies between the 0,2 and  1,1 , respectively. As shown in Fig. 2c) and Fig. 2d), a three quantum wires self-organized have the first and the second forms respectively. A five quantum wires which have a dome-like shape and a lens cross-section shape are shown in Fig. 2e) and Fig. 2f) respectively. The InAs/InP QW profiles in ɛ-space illustrated in Fig. 2 in the x, y  plane are and

are

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mapped in the β-space, the corresponding computational domain is u, v  plan. the selected domain size in the mapped space [23].

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The proposed method based on the coordinate transformation, which can fit the hetero interface of QW shown in Fig. 2 and simplify the computational domain as well as the resolution of Schrodinger equation in β-space, becomes possible compared to the case when the problem is solved in the original ɛ-space. It should be noted that the function given by

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relation (12) makes simpler the modelling of two shapes of quantum wires.

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In the next part, we use the general theoretical method proposed above to calculate the eigenvalues for the different shapes of quantum wire. In both directions, the discretisation

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steps are equal, i.e.   hu  hv . Therefore, the total number of grid points that corresponds to

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3. Results and discussion

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the size of the discretization matrix is ( N uv  N u  N v ) .

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In this part, We depict the numerical study of eigenvalues for InAs/InP quantum wires with different forms. The parameters used in our calculations are presented in table 1 and

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taken from Refs [24,25]. In fact, to characterize and explore our method, we analyze the influence of geometrical parameters of the quantum wire with different shape on the calculated eigenenergies. The analysis is based on the variation of the quantum wire height H. The effect of this parameter on the transition energies is also mentioned in the discussion.

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First, we have modelled an isolated quantum wire and calculated the ground state energies of electrons and heavy holes as a function of the height H for different models and for a fixed width of 14 nm. According to Fig. 3 we find that, for an isolated QW and all models, the energies decrease with the height H. As can be seen in Fig. 3a) the ground state energies for electron for model 1 is lower compared to model 2. For heavy holes, for H˂ 3.3 nm, the energies for 6

model 1 is higher than that for model 2, but for H ˃ 3.3 nm, the two models change the trend as illustrated in Fig. 3b). We have deduced the transition e-hh (electron-heavy hole) behaviour with the wire height H. The results are shown in Fig. 3c). The difference between the two models is of 2.22 meV with H=2.73 nm, 5.07 meV with H=3.22 nm and 8.28 meV with H=3.71 nm. To better explain the behaviour of the transition e-hh, we have plotted in Fig. 4 the

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transitions of energies as a function of the height for one, three and five QWs in the same figure.

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For model 1 and 2, the energy of 1 QW is a little bit higher compared to 3 QWs but

the difference between this energy and that obtained for 5 QWs is more remarkable. This difference remains nearly constant at about 7 meV and 8 meV for model 1 and 2, respectively. From these figures, we can note that the values for the transition energy of the isolated QW

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are always higher than the other two cases.

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By virtue of the importance of the energy levels, we have also studied the ground state

A

energies of electrons and heavy holes and the transition energies e-hh as a function of the

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number of quantum wires and for a fixed height of 4 nm. The results of this study are illustrated in Fig.5. As for electrons, it is observed that the energy decreases as the number of

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QWs increases. This means that the energy is divided with the increase of QWs number. In Fig. 5b), the energy of heavy holes remains constant with the increase of the QWs number. It is maximum and minimum at about 77.42 meV and 75.67 meV for models 1 and 2,

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respectively. As shown in Fig. 5c), for the two models, the transition energies decrease when the number of the wire increases.

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4. Conclusion

In the present work, we presented first an efficient method, which consists on a

A

coordinate transformation in order to model a quantum wire with different forms. The proposed method has been used to calculate the electron and hole energy levels of isolated and self-organized quantum wires. Thus, the Schrödinger equation is solved by the FDM in the new computational space. Then, we studied the influence of geometrical parameters on the shape of the quantum wire and we investigated the optical transition energies. According to the obtained results for the ground state energies for electrons and heavy holes as well as the 7

deduce transition energies e-hh, the geometrical parameters, the shape and the number of

A

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A

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quantum wires have an important role in the optical properties.

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References

[1] Z. Zanolli, M.E. Pistal, L.E. Frôberg, L. Samuelson, Condens. Matter 19 (2007) 295219. [2] V. Holovatsky, V. Gutsul, Condensed Matter Physics 10 (2007) 61-67.

(2013) 1-8.

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[4] M. Sabaeian, M. Shahzadeh, Physica E 68 (2015) 215-223.

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[3] W.K. Bae, Y.S. Park, J. Lim, D. Lee, L.A. Padilha, H. McDaniel, et al, Nat. Commun 4

[5] E. H. Sargent, Nat. Photon 6 (2012) 133-135.

[6] L. Ye, K.T. Yong, L. Liu, I. Roy, R. Hu, J. Zhu, et al, Nat. Nanotechnol 5 (2010) 391-400.

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[7] N. Ma, E.H. Sargent, S.O. Kelley, Nat. Nanotechnol 4 (2008) 121-125.

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[8] L.M. Herz, R.T. Phillips, Nat. Mater 1 (2002) 212-213.

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[9] R. Khordad, Superlatt. Microstruct, 54 (2013) 7-15.

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[10] H. Akiyana, M. Yoshita, L.N. Pfeiffer, K.W. West, A.Pinczuk, App. Phys. Lett, 82 (2003) 379-381.

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[11] M.Van den Broek, F.M. Peeters, Physica E 11 (2001) 345-355.

538-549.

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[12] R. Khordad, S. KheiryzadehKhaneghah, M. Masoumi, Superlatt. Microstruct, 47 (2010)

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[13] D.M. Gvozdic, A. Schalachetzki, J. App. Phys, 92 (2002) 2023-2034. [14] G. Creci, G.Weber, Semicond. Sci. Technol, 14 (1999) 690-694. [15] D. Kaufman, Y. Berk, B. Dwir, A. Rudra, A. Palevski, E. Kapon, Phys. Rev B, 59

A

(1999), 16.

[16] A. Gustafsson, F. Reinhardt, G. Biasiol, E. Kapon, App. Phys. Lett, 67 (1995) 36733675. [17] K. Chang, J.B. Xia, Phys. Rev B, 58 (1998) 2031-2037. [18] A. Gustafsson, F. Reinhardt, G. Biasiol, E. Kapon, App. Phys. Lett, 95 (1995) 3673. 9

[19] E. Levy, L. Sternfeld, M. Eshkol, M. Karpovski, B. Dwir, A. Rudra, E. Kapon, Y. Oreg, A. Palevski, Phys. Rev B 85 (2012) 045315. [20] Xiu-Zhi Duan, Guang-Xin Wang, De Liu, Bing-Ping Gou, Photonics and optoelectronics, 10 (2011) 5780629. [21] R. Cingolani, F. Sogawa, Y. Arakawa, R. Rinaldi, M. De Vittorio, A. Passaseo, A.

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Tourino, M. Catalano, L. Vasanelli, Phys. Rev. B 58 (1998) 1962-1966. [22] A. Bouazra, S. Abdi-Ben Nasrallah, M. Said, Physica E 75 (2016) 272-279.

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[23] A. Bouazra, S. Abdi-Ben Nasrallah, M. Said, Physica E 65 (2015) 93-99.

[24] P. Mohan, J. Motohisa, T. FuKui, Appl. Phys. Lett. 88 (2006) 133105-133107.

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[25] R.P. Schneider Jr, B.W. Wessels, J. Appl. Phys. 70 (1991) 405-408.

[26] D. Fuster, M.U. Gonzàlez, L. Gonzàlez, Y. Gonzàlez, T. Ben, A. Ponce, S.I. Molina,

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M

A

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Appl. Phys. Lett. 84 (2004) 23-25.

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Figure captions Figure 1: TEM picture of four stacked QWs, showing the InAs (dark) and the InP (bright) material [23]. Figure 2: The effects of parameters A a), D b), B c) and v d) on the shape of the quantum wire.

isolated QW (1QW) a) and b), three QWs c) and d) and five QWs e) and f).

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Figure 3: The profiles of quantum wire and the computational domain in the ɛ-space for an

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Figure 4: The ground state energies for electrons a) and heavy holes b). Transition energies ehh as a function of the height H for an isolated QW c).

Figure 5: The transitions energies e-hh as a function of the height H for model 1 a) and model

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2 b).

Figure 6: The ground state energies as a function of the number of QWs for electrons a) and

A

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M

A

heavy holes b). Transition energies e-hh as a function of the number of QWs c).

11

A ED

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12

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N

A

M

Figure 1 [26]

25

4

A=0.1 A=0.3 A=0.5

20

a)

D=0.05 D=0.15 D=0.25

3

b) y (nm)

y (nm)

15

10

2

1

0

0 -6

-4

-2

0

2

4

-6

6

-4

-2

0

x (nm)

2

4

6

4

4

3

3

d)

A

y (nm)

c) 2

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y (nm)

v=1.0 v=1.5 v=2.0

N

B=0.1 B=0.5 B=1.0

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x (nm)

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5

1

2

1

0

-6

-4

-2

0

2

4

-6

6

-4

-2

0

x (nm)

Figure 2

A

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x (nm)

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0

13

2

4

6

6

12 10

Model 2

Model 1

a)

4

v=0.0 v=2.0

8

2

4

InP

2

y(nm)

y(nm)

6

v=-1.0 v= 1.0

b)

InAs

0

InP InAs

0

-2

-2 -4

-6

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-4 -6

-6

-4

-2

0

2

4

6

-6

-4

-2

0

x(nm)

6

4

c)

v=0.0 v=2.0

v=-1.0 v= 1.0

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d)

2

y(nm)

4

N

y(nm)

6

2

-6 -20

-15

-10

-5

0

5

10

15

-6 -20

20

10

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12

-15

e)

0

5

Model 2 4

v=0.0 v=2.0

10

15

20

v=-1.0 v= 1.0

f)

2

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6

-5

6

Model 1

8

-10

x(nm)

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x(nm)

-4

M

-4

0

-2

A

0 -2

4

y(nm)

y(nm)

6

Model 2

Model 1

8

4

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12 10

2

x(nm)

2 0

0

-2

A

-2

-4

-4 -6

-30

-20

-10

0

10

20

30

x(nm)

-6 -30

-20

-10

0

x(nm)

Figure 3

14

10

20

30

0,24

0,115

Ground state energies for electrons

0,105

M1 M2

0,21

Energy (eV)

Energy (eV)

0,22

Ground state energies for heavy holes

0,110

a) 0,20 0,19

M1 M2

0,100 0,095

b)

0,090

IP T

0,23

0,085 0,080

0,18

0,075 2,4

2,6

2,8

3,0

3,2

3,4

3,6

3,8

4,0

SC R

0,17 4,2

2,4

Height H (nm)

2,6

U N

Transition e-hh

A

M

0,66 0,64

M1 M2

c)

0,62

ED

Transition energy (eV)

0,68

0,60 0,58

PT

3,0

3,2

3,4

Height H (nm)

0,70

2,4

2,6

2,8

3,0

3,2

3,4

Height H (nm)

Figure 4

A

CC E

2,8

15

3,6

3,8

4,0

4,2

3,6

3,8

4,0

4,2

0,68 0,68

Transition e-hh

Transition e-hh 0,66 1 QW 3 QWs 5 QWs

0,64

0,62

a)

0,60

1 QW 3 QWs 5 QWs

0,64

0,62

IP T

Transition energy (eV)

Transition energy (eV)

0,66

b) 0,60

0,58 0,56 2,4

2,6

2,8

3,0

3,2

3,4

3,6

3,8

4,0

2,4

4,2

2,6

2,8

3,0

3,2

3,4

Height H (nm)

A

CC E

PT

ED

M

A

N

Figure 5

U

Height H (nm)

SC R

0,58

16

3,6

3,8

4,0

4,2

0,195

0,0780

Ground state energies for electrons

Ground state energies for heavy holes

0,190

M1 M2

0,0770

0,175 0,170

a)

0,0765

0,0760

IP T

0,180

Energy (eV)

Energy (eV)

0,0775

M1 M2

0,185

b)

0,165

0,155

0,0750

1

2

3

4

5

SC R

0,0755

0,160

1

Number of QWs

2

U N A

0,594

M1 M2

M

0,592 0,590

Transition energies e-hh 0,586

ED

Energy (eV)

0,588

0,584 0,582 0,580

c)

PT

0,578 0,576

1

2

3 Number of QWs

Figure 6

A

CC E

3

Number of QWs

17

4

5

4

5

Table caption

Table 1: Values of the semiconductor physical parameters.

Parameter

520

(meV)

Electron effective mass in InP (

0.077

)

Heavy hole effective mass in InP ( Electron effective mass in InAs (

0.023

)

0.60

)

570

)

U

Energy gap in InAs (meV) (

0.85

)

Heavy hole effective mass in InAs (

)

A

CC E

PT

ED

M

A

N

Energy gap in InP (meV) (

IP T

Band offset

330

(meV)

SC R

Band offset

Value

18

1420