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Built-in electric field effects on donor bound excitons in wurtzite InGaN strained coupled quantum dots Yue-meng Chi, Jun-jie Shi ∗ State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, PR China Received 20 June 2006; received in revised form 1 September 2006; accepted 18 September 2006 Available online 27 September 2006 Communicated by R. Wu

Abstract Considering the three-dimensional confinement of the electrons and holes and the strong built-in electric field induced by the spontaneous and piezoelectric polarizations of the wurtzite Inx Ga1−x N/GaN strained coupled quantum dots (QDs), the positively charged donor bound exciton binding energy is calculated within the framework of the effective-mass approximation and variational method. Our results clearly indicate that the donor bound exciton binding energy sensitively depends on the donor position in the system, the structural parameters of the coupled QDs and the strong built-in electric field. The variation of this energy versus the donor position is in several tens of meV. © 2006 Elsevier B.V. All rights reserved. PACS: 73.21.La; 71.35.-y; 77.65.Ly; 77.84.Bw Keywords: InGaN coupled quantum dots; Donor bound exciton; Built-in electric field; Exciton binding energy

1. Introduction In recent years, wide-band gap wurtzite III–V nitrides, such as InN, GaN, AlN and their ternary compounds have attracted much attention due to potential device applications. For instance, they are widely used in the fabrication of optoelectronic devices: blue/ultraviolet lasers, laser diodes (LDs), light emitting diodes (LEDs), and ultraviolet detectors because of their large and direct band gaps [1–3]. Most of these devices contain InGaN/GaN quantum wells (QWs) as the active region. It is well known that many nanometer-scale In-rich islands called In-rich quantum dots (QDs) are spontaneously formed within the InGaN active layers in wurtzite Inx Ga1−x N/GaN semiconductor heterostructures [4]. The three-dimensional (3D) confinement of the electrons and holes in the In-rich QDs has significant technological applications, such as lasers, charge-storage devices, and all-optical quantum information processors because of stronger band-to-band emission [5]. Considering strain of the wurtzite Inx Ga1−x N/GaN heterostructure due to large lattice mismatch (11%) between Inx Ga1−x N and GaN [6], the spontaneous and piezoelectric polarizations of the structure [7] give rise to large polarization sheet charges on the interfaces or surfaces of the system, which, in turn, create a definite internal electric field inside the heterostructure [8]. For the commonly used [0001]-oriented wurtzite Inx Ga1−x N/GaN strained QWs, the magnitude of the built-in electric field is known as in the order of MV/cm [8–10]. Such a strong field will bring about a remarkable reduction of the effective band gap of QWs or QDs. It will also cause a distinct spatial separation of the electrons and holes in the coupled QD structure because the electrons and holes are localized in the two different QDs due to the built-in electric field (please refer to Fig. 2 of Ref. [9]). Hence the exciton states and the optical properties of the Inx Ga1−x N/GaN strained QDs are obviously influenced by the 3D confinement of the electrons and holes and the strong built-in electric field.

* Corresponding author.

E-mail addresses: [email protected] (Y.-m. Chi), [email protected] (J.-j. Shi). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.09.038

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Fig. 1. A diagram of a cylindrical Inx Ga1−x N/GaN coupled QD structure with radius R, surrounded by two large energy gap materials Iny Ga1−y N (y < x) in the radial direction and GaN in the z-direction. The heterointerfaces are located at z = z0 , z1 , z2 and z3 . Here, nine black dots stand for nine different donor positions: zd = −1.0 + z0 , z0 , (z0 + z1 )/2, z1 , (z1 + z2 )/2, z2 , (z2 + z3 )/2, z3 , and 1.0 + z3 nm, which are denoted as position I, II, III, IV, V, VI, VII, VIII, and IX in the text. For simplicity, we take LInGaN = z1 − z0 = z3 − z2 , LGaN = z2 − z1 in our calculations.

The confined exciton states in wurtzite Inx Ga1−x N single QDs have been solved, including the effects of piezoelectric and spontaneous polarizations [10,11]. Considering coupled multiple QD structures are widely adopted in the commonly used GaN based optoelectronic devices, such as LEDs and LDs [1], the theory of Refs. [10,11] has been extended to the case of the Inx Ga1−x N/GaN coupled QDs recently [9,12]. The electrons and holes can be localized in different QDs of the coupled QD structure and the spatial indirect excitons will thus be formed. The exciton state in a coupled QD structure becomes much more complex than that in a single QD system. Moreover, we know that the nitride-based nanostructures are defect dominated materials [13]. Excitons can also be bound to point-defects, for example, the metal vacancy VM and the nitrogen vacancy VN , to form bound exciton complexes [14], such as a neutral (an ionized) bound exciton composed of an exciton combined with a neutral (an ionized) donor impurity denoted as D0 X (D+ X). The Hamiltonian of these bound exciton complexes is different from that of the so-called excitonic complexes or trions, for example, the positively charged exciton X+ due to one electron combined with two holes, and the negatively charged exciton (X− ) formed by the two electrons and one hole [14]. For the difference between X− and D+ X, please refer to Eq. (1) of Ref. [15] and Eq. (1) of Ref. [16]. Recently, the optical transitions arise from D0 X and D+ X in GaN epitaxial layers have been discovered at low temperature (T < 50 K) [17–19]. Furthermore, the PL spectra of GaN are dominated by the transitions due to D0 X and D+ X as T < 20 K [17,18]. Consequently, it is necessary and imperative to investigate the donor bound exciton states and related interband optical transitions in the nitride-based coupled QDs. Quite recently, the donor bound excitons in wurtzite InGaN single QDs have been studied by Shi and Tansley [20]. To the best of our knowledge, calculations of the donor bound exciton states in Inx Ga1−x N/GaN coupled QDs due to the effects of the strong built-in electric field are still absent at present. In this Letter, we pay attention to the donor bound exciton D+ X states in a wurtzite Inx Ga1−x N/GaN symmetric cylindrical coupled QD system. Our primary aim here is to calculate ionized donor bound exciton binding energy as functions of the structural parameters of the QDs, the donor position, and the built-in electric field by utilizing a variational method within the framework of the effective-mass approximation. Realistic cases, including the donor in the QDs and in the surrounding barriers, are considered. 2. Theory Let us now take into account an ionized donor bound exciton D+ X confined in a symmetric cylindrical wurtzite Inx Ga1−x N/GaN strained coupled QD structure, in which different ionized donor positions in the QDs and barriers are considered. The QDs are surrounded by two large energy gap materials Iny Ga1−y N (y < x) in the radial direction and GaN barrier layer in the z-direction (see Fig. 1). The cylindrical coupled QDs are indexed by their radius R, the barrier thickness LGaN between the two coupled Inx Ga1−x N QDs and the QD height LInGaN . Based on the effective-mass approximation, the Hamiltonian of the bound exciton D+ X can be written as e2 1 1 1 Hˆ D+ X = Hˆ e + Hˆ h + Eg − (1) − + , 4πε0 ε¯ |re − rd | |rh − rd | |re − rh |

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where Hˆ e (Hˆ h ) stands for the electron (hole) Hamiltonian, Eg indicates the band gap energy of the Inx Ga1−x N material, re , rh and rd are the position vectors of the electron, hole and donor, respectively. The notation ε0 denotes the permittivity of free space. For simplicity, we introduce ε¯ as the effective mean relative dielectric constant of the embedding material between the electron and hole. The physical reason has been given in Ref. [21]. In this Letter, we assume ε¯ = εInx Ga1−x N for all regions of the heterostructure. This is a reasonably good approximation since the electron and hole are to a large extent confined in the Inx Ga1−x N QDs due to the large barrier heights. Moreover, we choose the conduction band bottom and the valence band top of the Inx Ga1−x N layer at the interface z = z0 as the reference energy level for the electron in the conduction band and the hole in the valence band, respectively. The Hamiltonian of the electron (hole) in the cylindrical coordinates reads ∂ h¯ 2 1 ∂ ∂2 1 ∂2 ρj + 2 2 + 2 + V (ρj , zj ) + Vj (zj ), Hˆ j = − ∗ (2) 2mj ρj ∂ρj ∂ρj ρj ∂ϕj ∂zj where the subscript j = e or h denotes the electron or hole. The electron (hole) effective mass m∗j is notated as ∗ mIj z0 < zj < z1 or z2 < zj < z3 , ∗ mj = m∗IIj otherwise.

(3)

It is interesting to point out that, compared with the case of a constant mass, a material-dependent effective mass will generally have a large influence on the electron (hole) energy levels in a semiconductor QW, especially for the narrow QW with the well width less than 4 nm [22]. In order to obtain the exact confined energy for the electron and hole in the Inx Ga1−x N coupled QDs, we explicitly include the electron (hole) material-dependent effective mass here as suggested by Harrison [22]. The electron (hole) confinement potential in the strained coupled QD structure is given by V (ρj ), z0 < zj < z1 or z2 < zj < z3 , V (ρj , zj ) = (4) vIIj , otherwise, 0, ρj R, V (ρj ) = (5) vIj , ρj > R. For the Inx Ga1−x N/GaN strained coupled QD structure shown in Fig. 1, we choose the electron and hole confinement potentials in Eqs. (4) and (5) as vIj = pj {1.5(x − y) + 3.2[x(1 − x) − y(1 − y)]} eV and vIIj = pj [1.5x + 3.2x(1 − x)] eV with pe = 0.6 and ph = 0.4 [11]. The static electric potential resulted from the built-in electric fields can be expressed as ⎧ 0, zj z0 , ⎪ ⎪ InGaN (z − z ), ⎪ z ±eF ⎨ j 0 0 < zj < z1 , InGaN (z − z ) + eF GaN (z − z )], z z z , ±[eF Vj (zj ) = (6) 1 0 j 1 1 j 2 ⎪ ⎪ z2 < zj < z3 , ⎪ ±eF InGaN (zj − z3 ), ⎩ 0, zj z3 , where the sign − (+) is for the electron (hole). The values of the electric field F InGaN in the Inx Ga1−x N layers and F GaN in the GaN barrier layer between the two coupled QDs are given in Ref. [9]. If we assume that the in-plane and on-axis motions of the electron (hole) are weakly coupled, as has been done in Refs. [23,24], the wave function of the electron (hole) confined in the Inx Ga1−x N coupled QDs can be approximately written as ψj (ρj , ϕj , zj ) = f (ρj )h(zj )eimϕj ,

m = 0, ±1, ±2, . . . ,

(7)

where m is the electron (hole) z-component angular momentum quantum number. The radial wave function f (ρj ) and the corresponding confinement energy equation of the electron (hole) can be obtained by using the m-order Bessel function Jm and the modified Bessel function Km . For the z-axis motion of the electron (hole), the wave function h(zj ) and the related confinement energy equation can be expressed by means of the Airy functions Ai and Bi. The variational method is used to calculate the bound exciton ground state energy. Considering the correlation of the electron– hole relative motion, the trial wave function can be chosen as, ΦD+ X (re , rh ) = ψe (ρe , ϕe , ze )ψh (ρh , ϕh , zh )e−αρeh e−βzeh , 2

2

(8)

= (xe − xh + where ψe (ψh ) is the electron (hole) wave function confined in the Inx Ga1−x N/GaN coupled QD structure, (ye − yh )2 and zeh = ze − zh . The variational parameter α is responsible for the in-plane correlation, and β for the correlation of the relative motion in the z-direction. It is worthwhile to point out that, just for the sake of simplifying our calculations, the simple Gaussian function Eq. (8), which has also been widely adopted in literature [23,25], is chosen to describe the correlation of the electron–hole relative motion. An improving choice can be expected by means of an expansion in a series of Gaussians [26] or by using the hydrogen-type trial wave function [27]. 2 ρeh

)2

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Fig. 2. Variation of Eb versus the barrier thickness, for eight different donor positions, taking the strong built-in electric field into account. Additionally, the solid line refers to the free exciton binding energy without the donor. The dot–dot-dashed line indicates Eb without the built-in electric field as the donor is placed in the position III.

The ground state energy of the donor bound exciton in our Inx Ga1−x N/GaN strained coupled QDs can be determined by ED+ X = min α,β

ΦD+ X |Hˆ D+ X |ΦD+ X . ΦD+ X |ΦD+ X

(9)

The bound exciton binding energy Eb can be defined as E b ≡ E e + E h + Eg − E D+ X ,

(10)

where Ee (Eh ) is the electron (hole) confinement energy in the wurtzite Inx Ga1−x N/GaN strained coupled QDs. 3. Numerical results and discussion Considering the different donor positions and the strong built-in electric field effects, the donor bound exciton binding energy as functions of the structural parameters of the Inx Ga1−x N/GaN strained coupled QDs, such as the barrier thickness LGaN , the QD height LInGaN , the In fraction x, and the radius R, is calculated by using the aforementioned theory. All material parameters used in our calculations are the same as in Ref. [11]. The donor bound exciton binding energy Eb as a function of the barrier thickness LGaN between the two QDs is shown in Fig. 2. Our calculations indicate that the electron is located in the left QD of the structure due to the strong built-in electric field. Moreover, the hole is more strongly localized in the right QD due to its larger effective mass than the electron (please refer to Fig. 2 of Ref. [9]). We can find from Fig. 2 that Eb is remarkably influenced by the donor position. Fig. 2 shows that the binding energy corresponding to the donor in the left QD is several tens of meV larger than that in the right QD. It is because the electron is localized in the left QD, and the strong Coulomb attractive interaction between the ionized positively charged donor and electron enhances the value of Eb . On the other hand, the strong Coulomb repulsive interaction between the donor and hole reduces Eb if the donor is placed in the right QD. It is more interesting to note that the donor bound exciton states can be approximately described by the free exciton states if the donor is placed in the middle of the barrier between the two QDs. The physical reason is that the Coulomb interaction intensity between the electron (hole) and the donor becomes approximately equal. Furthermore, Fig. 2 shows that Eb is basically invariable with increase of the barrier width LGaN when the donor is located in the positions: I, II and III (see Fig. 1). In these cases, the bound exciton states are mainly determined by the donor–electron interaction. This is because the hole is strongly localized in the right QD, and the interaction between the hole and donor is weak due to their large spatial separation. Hence Eb is basically unchangeable when LGaN increases. Moreover, as the donor is placed on the right interface (zd = z1 ) of the left QD, Eb increases smoothly with increase of LGaN . It is because the repulsive interaction between the ionized donor and hole is reduced due to increase of their relative distance. In order to know the influence of the built-in electric field on the donor binding energy, we have also calculated Eb as a function of the barrier thickness LGaN when fixing the donor in the position III without the built-in electric field. Our results show that, considering the strong built-in electric field, Eb is approximately 30 meV larger than that without the field. This is because the electron (hole) wave function is localized in the two

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Fig. 3. Variation of Eb versus the QD height, for seven different donor positions, considering the strong built-in electric field effects. Besides, the solid line refers to the free exciton binding energy without the donor.

QDs with equal probability if ignoring the field. Comparing with the situation of the built-in electric field, the Coulomb interaction of the electron (hole) and donor is reduced (enhanced) as the donor is placed in the left QD without the field. We can also see from Fig. 2 that Eb monotonically decreases with increase of LGaN when the donor is placed in the vicinity of the right QD. The physical reason is that the Coulomb attractive interaction between the electron and donor is reduced when the barrier width increases. It is worthwhile to point out that the bound exciton states cannot be formed when the donor is in the middle of the right QD due to the strong Coulomb repulsive interaction between the ionized donor and hole. For the current coupled QD structure, we find that the bound exciton states become unstable if LGaN > 1.4, 1.6 and 2.9 nm when the donor is placed in the positions: VIII, VI and IX, respectively. This is due to large spatial separation between the hole (donor) and the electron and the strong repulsive interaction between the hole and the ionized positively charged donor located in the right QD. Based on our above analyses, it can be concluded that the donor bound exciton binding energy is strongly influenced by the barrier width LGaN , the donor position, and the strong built-in electric field. In Fig. 3, we calculate the donor bound exciton binding energy Eb as a function of the QD height LInGaN , in which different donor positions are included. First, we find Eb decreases monotonically with increase of LInGaN if the donor is placed in the positions: I and II. This is because the Coulomb attractive interaction between the electron and donor is reduced due to increase of their distance. Moreover, Fig. 3 shows that Eb has a complicated relation with increase of LInGaN if the donor is placed in the positions: III and IV. The value of Eb slightly decreases and then increases obviously as LInGaN increases. The physical reason can be understood based on the calculation of the built-in electric field in the structure (see Fig. 1(c) of Ref. [9]). With increase of LInGaN , the electron–hole relative distance in the z-direction is enlarged due to decrease of the built-in electric field in the QDs. Hence the value of Eb slightly decreases. When LInGaN continuously increases (LInGaN > 2.6 nm), the electron will be strongly localized in the left QD due to increase of the effective height of the barrier between the two QDs. The Coulomb interaction between the electron and donor will thus dominate the donor bound exciton states. Fig. 3 also indicates that the donor bound exciton states can be roughly described by the free exciton states if the donor is placed in the middle of the barrier between the two QDs. The reason has been analyzed in Fig. 2. We can also see from Fig. 3 that Eb hardly varies when the donor is placed in the position IX. This is because increase of the QD height does not remarkably influence the localization of the hole wave function. Moreover, Fig. 3 also shows that the bound exciton will become unstable when LInGaN < 2.7 nm if the donor is placed in the position VIII. The reason is that the strong Coulomb repulsive interaction between the donor and hole inhibits the formation of the bound exciton. When the LInGaN > 2.7 nm, spatial separation between the donor and hole becomes large. Hence a stable bound exciton state can exist and the corresponding values of Eb increase slightly. Additionally, the bound exciton states cannot be formed when the donor is placed in the positions: VI and VII. This is because the hole is localized near the left interface of the right QD, where the repulsion between the donor and hole becomes larger. We can thus obtain the conclusion that the QD height has an important impact on the donor bound exciton states in our Inx Ga1−x N/GaN strained coupled QD structure. The exciton binding energy Eb as a function of the QD height LInGaN and barrier thickness LGaN is shown in Fig. 4 for the case of the donor in the position III. We can find that Eb is strongly influenced by LInGaN and LGaN . The variation of Eb is in several tens of meV. This has been explained in the above. We can also see from Fig. 4 that the influence of LInGaN and LGaN on Eb becomes

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Fig. 4. Dependence of Eb on the QD height and barrier thickness, for the donor in the position III of the In0.2 Ga0.8 N/GaN coupled QDs.

Fig. 5. Variation of Eb versus In fraction x. Here the donor is placed in the position III.

much more obvious when LGaN and LInGaN are large. The physical reason is as follows. The Coulomb interaction between the electron and donor dominates the donor bound exciton states if the barrier thickness and the QD height increase. In Fig. 5, we investigate the donor bound exciton binding energy Eb as a function of the In composition x when the donor is placed in the position III. We find that Eb increases monotonically if x increases. This is because the potential well of the electron (hole) becomes much deeper with increase of the In composition x. Hence the electron wave function is more strongly localized inside the left QD and the Coulomb interaction between the electron and the ionized donor is enhanced if x increases. The bound exciton binding energy as a function of the cylindrical radius R is shown in Fig. 6. We can see from Fig. 6 that Eb decreases monotonically if R increases. This is due to increase of the electron–hole in-plane relative distance and weakening of the Coulomb interaction when R increases. 4. Conclusions Considering the strong built-in electric field effects, the donor bound exciton binding energy in wurtzite Inx Ga1−x N/GaN strained coupled QDs is calculated within the framework of the effective mass approximation and variational method. Our results indicate that the structural parameters (the barrier thickness LGaN , the QD height LInGaN , the In composition x, and the radius R) of the coupled QDs, the donor position, and the strength of the built-in electric field have a remarkable influence on the exciton states.

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Fig. 6. Dependence of Eb on the radius R. Here the donor is fixed in the position III.

The donor bound exciton binding energy can be obviously modified by the built-in electric field in the order of several tens of meV. Generally, the donor binding energy with the donor in the left QD is always larger than that in the right QD. The bound exciton states can be approximately treated as the free exciton states when the donor is in the middle of the barrier between the two QDs. If the donor is placed in some special positions of the right QD, the bound exciton states can become unstable with increase (decrease) of LGaN (LInGaN ). The present results are useful for design of some important blue/ultraviolet LDs, LEDs, and related photoelectric devices constructed by using wurtzite Inx Ga1−x N/GaN heterostructures. Experimental results for the bound exciton states in the current Inx Ga1−x N/GaN strained coupled QD structure are still lacking at present. We hope that our theory can stimulate further investigations of the physics, and device applications of group-III nitrides. Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant Nos. 60276004 and 60390073. References [1] S. Nakamura, G. Fasol, The Blue Laser Diode, Springer-Verlag, Berlin, 1997. [2] A. Bell, I. Harrison, T.S. Cheng, D. Korakakis, C.T. Foxon, S. Novikov, B.Y. Ber, Y.A. Kudriavtsev, Semicond. Sci. Technol. 15 (2000) 789. [3] S. Nakamura, M. Senoh, S. Nagahama, T. Matsushita, H. Kiyoku, Y. Sugimoto, T. Kozaki, H. Umemoto, M. Sano, T. Mukai, Jpn. J. Appl. Phys. 38 (1999) L226, Part 2. [4] J. Kim, K. Samiee, J.O. White, J.-M. Myoung, K. Kim, Appl. Phys. Lett. 80 (2002) 989. [5] Y. Masumoto, T. Takagahara, Semiconductor Quantum Dots, Springer-Verlag, Berlin, 2002, p. 1805. [6] P. Waltereit, M.D. Craven, S.P. DenBaars, J.S. Speck, J. Appl. Phys. 92 (2002) 456. [7] F. Bernardini, V. Fiorentini, D. Vanderbilt, Phys. Rev. Lett. 79 (1997) 3958. [8] O. Mayrock, H.J. Wünsche, F. Henneberger, Phys. Rev. B 62 (2000) 16870. [9] J.J. Shi, C.X. Xia, S.Y. Wei, Z.X. Liu, J. Appl. Phys. 97 (2005) 083705. [10] J.J. Shi, Solid State Commun. 124 (2002) 341. [11] J.J. Shi, Z.Z. Gan, J. Appl. Phys. 94 (2003) 407. [12] C.X. Xia, S.Y. Wei, Phys. Lett. A 346 (2005) 227. [13] K.S.A. Butcher, M. Wintrebert-Fouquet, P.P.-T. Chen, T.L. Tansley, H. Dou, S.K. Shrestha, H. Timmers, M. Kuball, K.E. Prince, J.E. Bradby, J. Appl. Phys. 95 (2004) 6124. [14] C.F. Klingshirn, Semiconductor Optics, Springer-Verlag, Berlin, 1997. [15] E. Anisimovas, F.M. Peeters, Phys. Rev. B 68 (2003) 115310. [16] I.C. da Cunha Lima, A. Ghazali, P.D. Emmel, Phys. Rev. B 54 (1996) 13996. [17] A.K. Viswanath, J.I. Lee, C.R. Lee, J.Y. Leem, D. Kim, Appl. Phys. A 67 (1998) 551. [18] D.C. Reynolds, D.C. Look, B. Jogai, V.M. Phanse, R.P. Vaudo, Solid State Commun. 103 (1997) 533. [19] D.C. Look, G.C. Farlow, P.J. Drevinsky, D.F. Bliss, J.R. Sizelove, Appl. Phys. Lett. 83 (2003) 3525. [20] J.J. Shi, T.L. Tansley, Solid State Commun. 138 (2006) 26. [21] D. Schooss, A. Mews, A. Eychmüller, H. Weller, Phys. Rev. B 49 (1994) 17072. [22] P. Harrison, Quantum Wells, Wires and Dots, Wiley, Chichester, 2005. [23] S.L. Goff, B. Stébé, Phys. Rev. B 47 (1993) 1383.

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