Polaron effects due to interface optical-phonons ... - Wiley Online Library

phys. stat. sol. (b) 242, No. 5, 1010 – 1021 (2005) / DOI 10.1002/pssb.200402137

Polaron effects due to interface optical-phonons in wurtzite GaN/AlN quantum wells Yao-hui Zhu and Jun-jie Shi* State Key Laboratory for Mesoscopic Physics, and School of Physics, Peking University, Beijing 100871, People’s Republic of China Received 14 August 2004, revised 28 October 2004, accepted 24 November 2004 Published online 2 February 2005 PACS 63.20.Dj, 63.20.Kr, 63.22.+m, 71.38.–k, 78.67.De Considering the effects of the built-in electric field (BEF) induced by the spontaneous and piezoelectric polarizations of wurtzite GaN/AlN quantum wells (QW’s), the polaron energy shift and the effective mass due to the electron interactions with the interface optical-phonons are investigated theoretically by means of Lee-Low-Pines variational approach. We find that the BEF has a remarkable influence on the polaron effects especially for a QW with well width d > 6 nm. The polaron energy shift increases slowly and its effective mass approaches to a constant if d is further increased. On the contrary, both the polaron energy shift and the effective mass decrease slowly with the increasing of d if the BEF is ignored. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1

Introduction

In recent years, the investigation of the optical, electronic and lattice dynamical properties of quantum heterostructures based on the wide-band-gap group-III nitrides GaN, AlN, InN and their ternary compounds AlxGa1–xN and InxGa1–xN has attracted much attention due to their potential device applications, particularly for high-brightness blue/green light emitting diodes (LED’s) and laser diodes (LD’s) [1 –15]. The group-III nitrides usually crystallize in wurtzite structure. Their fundamental physical properties can be largely affected and even determined by the spatial quantization of the electron states, the anisotropy of the crystal structures and the strains of the heterostructures. Hence the GaN-based quantum heterostructures become quite attractive from both device applications and purely physical viewpoint. The group-III nitrides are polar semiconductors. It is well known that an electron moving slowly in a heterostructure of polar semiconductors may cause a distortion of the lattice, establishing a polarization field which acts back on the electron whose properties are then modified; in particular, the electron acquires a self-energy and an enhancement of its Bloch effective mass. The single electron, together with its accompanying distortion, is called a polaron. The polaron effects can strongly influence the optical and transport properties of polar semiconductor heterostructures. Hence the polaron has been a major topic of great interest in quantum theory of solids. The understanding of lattice dynamics and polaron effects in anisotropic wurtzite quantum heterostructures is primitive, which has not only important theoretical meaning but also practical significance for device applications. It has been known that the anisotropy of the dielectric properties of a polar crystal and of the effective band mass influences the energy and mass of polarons. Considering both the anisotropy of the effective mass and the static and high frequency dielectric constants, polarons in anisotropic uniaxial crystals have been studied for many years [16– 22]. It is found that both the polaron selfenergy and the effective mass can decrease or increase depending on the anisotropic parameters, i.e., the lattice dielectric functions e ^ *

Corresponding author: e-mail: [email protected], Phone: +86-10-62757594, Fax: +86-10-62751615 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original Paper

phys. stat. sol. (b) 242, No. 5 (2005) / www.pss-b.com

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and e z in the perpendicular direction and the z -direction along the c -axis of the bulk materials, and the effective mass m^ and mz . The anisotropy in the effective mass can also be reduced by the electron– phonon interactions [21]. Compared with the anisotropic bulk materials, the polaron effects in anisotropic polar semiconductor quantum heterostructures are markedly different due to the presence of heterointerfaces and the lattice mismatch. Generally, the electron will be localized in the quantum well (QW) layer which leads to the spatial quantization of the electron states. The phonon modes become much more complicated because of the presence of heterointerfaces and the anisotropy of the crystals. The interface optical phonon modes and the propagating modes, together with the quasiconfined modes, the exactly confined modes, and the half-space modes, were found to coexist in a wurtzite quasi-twodimensional (Q2D) multilayer heterostructure [10 –12]. Moreover, the zone-center phonon frequencies are shifted to higher values owing to the strains of the heterostructures [10, 11]. A strong built-in electric field (BEF) in the order of MV/cm can be induced by the piezoelectric and spontaneous poalrization in wurtzite GaN-based QW’s [9, 23]. Hence the polaron effects in wurtzite group-III nitrides QW’s are much more complicated than the situation in anisotropic bulk materials and need to be studied deeply. The polaron in GaN material is large polaron with its radius approximately 9 nm, which is larger than the lattice constant of the bulk GaN [5.185 (3.189) Å along the parallel (perpendicular) direction of the c -axis]. It can thus be described by Fröhlich electron–phonon interaction Hamiltonian in which the crystal lattice is treated as continuum dielectric. Recently, the effective mass of a Q2D polaron confined in the GaN/AlGaN heterointerface is measured by experiments [24 –26] and calculated theoretically [25, 27]. In Refs. [25, 27], the phonon modes were simply replaced by the ones of the bulk materials. The polaron binding energy and the effective mass of bulk III –V nitride compounds with wurtzite structure were also investigated by means of various perturbation methods in Refs. [28 –32]. However, as we know, the electron –phonon coupling constant in the bulk GaN material is estimated as a GaN = 0.49 [27], which is in the intermediate coupling range. Hence the Lee-Low-Pines (LLP) variational method [33] can be expected to give a better result than the perturbation methods due to strong electron-phonon interactions in GaN-based quantum heterostructures. Recently, Shi [10, 11] solved the interface optical-phonon (IOP) modes in an arbitrary wurtzite Q2D multilayer heterostructure based on the dielectric-continuum (DC) model [34] and Loudon’s uniaxial crystal model [35, 36]. The electron-interface-phonon interaction Fröhlich-like Hamiltonian in wurtzite Q2D multiple quantum wells (MQW’s) or superlattices (SL’s) have also been derived and discussed in Refs. [10, 11]. To the best of our knowledge, the polaron effects in important GaN-based QW’s and SL’s have not been investigated in depth up to now. The purpose of the present article is devoted to theoretical study of polaron effects due to electron-IOP interactions in GaN/AlN QW’s by means of the LLP variational method, in which the strong BEF due to piezoelectricity and spontaneous polarization and the anisotropy of the wurtzite crystal structures and the strains of the QW’s are included. It is more interesting to point out that the polaron effects due to electron-IOP interactions are important because the wave functions of the electron confined in GaN-based QW’s will be pushed close to the one interface due to the strong BEF [9]. The paper is organized as follows. In Section 2, the Fröhlich polaron Hamiltonian due to the electronIOP interactions is given first. The polaron energy shift and the effective mass are then calculated theoretically. In Section 3, we present and discuss our numerical results. Finally, the main conclusions obtained in this paper are summarized in Section 4.

2

Theory

2.1 Hamiltonian We consider a GaN/AlN single QW. The two heterointerfaces are located at z = - d / 2 and d/ 2 , respectively. Here, we take the z -axis (c -axis) to be perpendicular to the heterointerfaces. Within the framework of the effective mass approximation, the total Hamiltonian for the coupling of an electron to the IOP’s can be written as H = H e + H IO + H e - IO .

(1) © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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Yao-hui Zhu and Jun-jie Shi: Polaron effects due to interface optical-phonons

The first term is the Hamiltonian of an electron confined in a potential well V ( z ) in the z -direction and is given by He =

2 2 pˆ x + pˆ y

2 m^

+

2 pˆ z - eF ( z ) z + V ( z ) , 2 mz

(2)

where the electron effective mass is given as Ïm^,GaN , |z| < d / 2, m^ = Ì Ó m^,AlN , |z| > d / 2,

(3)

Ïmz ,GaN , |z| < d / 2, mz = Ì Ó mz ,AlN , |z| > d / 2,

(4)

and

Here the subscripts ^ and z denote the perpendicular direction and the parallel direction of the z-axis. In Eq. (2), F is the BEF caused by the polarization of the QW structure. In general, the direction of the electric field F depends on the orientation of the piezoelectricity and spontaneous polarization and can be determined by both the polarity of the crystal and the strains of the QW structure. For definiteness, we choose the positive direction of F opposite to the z -direction and F ( z ) can be written as [8, 9], Ï F , |z| < d / 2, F (z) = Ì Ó 0, |z| > d / 2.

(5)

The confinement potential V ( z) is given by Ï 0, |z| < d / 2, V (z) = Ì ÓV0 , |z| > d / 2,

(6)

where V0 is the conduction band offset between GaN and AlN layers. According to Eq. (2.42) of Ref. [10], the last two terms in Eq. (1) can be written as H IO = Â Â w j (q^ ) ÈÎ aˆ †j ( q^ ) aˆ j ( q^ ) + 12 ˘˚ ,

(7)

H e - IO = Â Â e iq^ ◊r G j (q^ , z ) [ aˆ j ( q^ ) + aˆ †j ( - q^ )] ,

(8)

q^

j

and j

q^

where w j (q^ ) represents the frequency of the j th IOP mode with the wavevector q^ and can be obtained by solving Eq. (2.28) of Ref. [10] numerically. The electron–phonon coupling function G j (q^ , z ) describes the coupling strength of a single electron located at the position z with the j -th IOP mode and is given by Eq. (2.43) of Ref. [10]. 2.2 Polaron energy shift and effective mass In this subsection, we will investigate the polaron energy and the effective mass in wurtzite GaN/AlN QW’s by using the LLP variational method. Following Ref. [33], we can introduce the two unitary transformation operators as follows, Ï È ˘ ¸ Sˆ = exp Ìi Í K ^ - Â Â q^ aˆ †j ( q^ ) aˆ j ( q^ ) ˙ ◊ r ˝ , j q^ ˚ ˛ Ó Î © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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and Ï ¸ (10) Uˆ = exp ÌÂ Â [ aˆ †j ( q^ ) f j ( q^ , z ) - aˆ j ( q^ ) f j* ( q^ , z )] ˝ , Ó j q^ ˛ where r is the electron position vector in the xy plane, and f j ( q^ , z) is a variational parameter. Generally, the function f j ( q^ , z ) should depend on the coordinate z [37 –39]. This is because the conservation of the total momentum is restricted to the components parallel to the heterointerfaces. Hence the problem becomes much more complicated. Considering the strong BEF (in MV/cm) caused by polarization of the QW structure, the electron is highly localized in the vicinity of one interface of deep GaN QW’s with the well depth approximately 2.1 eV (see Fig. 2), f j ( q^ , z) becomes weekly dependent on the coordinate z . For simplicity, we thus ignore the dependence of f j ( q^ , z) on z and let f j ( q^ , z) ª f j ( q^ ) in the present paper. In Eq. (18), K ^ is the eigenvalue of the total momentum operator Kˆ ^ of the polaron in the xy plane, which can be written as Kˆ = pˆ + q aˆ † ( q ) aˆ ( q ) , (11) ^

ÂÂ

^

j

^

j

^

^

j

q^

where pˆ ^ is the momentum operator of the bare electron in the xy plane. After carrying out the two unitary transformations, we have ˆˆ. H Æ H = Uˆ -1Sˆ -1 HSU (12) Let us now consider a slow electron with p^ ª 0 and neglect the zero point energy of the phonons, we can obtain, 2 pˆ 2 K ^2 + z - eF ( z ) z + V ( z ) H = 2 m^ 2 m z

2 q^2 2 K ^ ◊ q^ ˘ † È + Â Â Í w j (q^ ) + ˙ ÎÈ aˆ j ( q^ ) + f j* ( q^ ) ˚˘ [ aˆ j ( q^ ) + f j ( q^ )] m^ 2 m^ Î ˚ j q^ † + Â Â G j (q^ , z ) ÈÎ aˆ j ( q^ ) + f j ( q^ ) + aˆ j ( q^ ) + f j* ( q^ ) ˘˚ q^

j

+Â

Â

j , j ¢ q^ ,q^¢

2 q^ ◊ q ¢ ^ † [ aˆ j ¢ ( q ¢ ^ ) + f j*¢ ( q ¢ ^ )] [ aˆ †j ( q^ ) + f j* ( q^ )] 2 m^

[ aˆ j ¢ ( q ¢ ^ ) + f j ¢ ( q ¢ ^ )] [ aˆ j ( q^ ) + f j ( q^ )] .

(13)

The trial wave function of H can be chosen as |Y Ò =

1 A

e iK ^ ◊ r h ( z ) | 0 Ò ,

where | 0Ò is the phonon vacuum state,

(14)

1

eiK^ ◊r is the normalized eigenfunction of the electron in the xy

A plane and A is the cross-sectional area of the heterostructure, and h(z) is the eigenfuntion of the electron in the z direction and satisfies the following equation È 2 d 2 ˘ Í - 2 m dz 2 - eF ( z ) z + V ( z ) ˙ h( z ) = Ez h ( z ) . Î ˚ z

(15)

After solving the above equation, we obtain ÏC[Bi (t L ) - l Ai (t L )] e kL ( z + 2 ) , z £ - d2 , Ô h( z ) = ÌC[Bi (t ) - l Ai (t )] , |z| £ d2 , d Ô kR ( 2 - z ) , z ≥ d2 , ÓC[Bi (t R ) - l Ai (t R )] e d

(16)

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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where l∫

KBi ¢ (t L ) + kL Bi (t L ) , KAi ¢ (t L ) + kL Ai (t L )

K∫

mz ,AlN Ê 2 mz ,GaN eF ˆ 3 Á ˜¯ , 2 mz ,GaN Ë

(18)

k L2 ∫

2 mz ,AlN (V0 + deF / 2 - Ez ) , 2

(19)

k R2 ∫

2 mz ,AlN (V0 - deF / 2 - Ez ) , 2

(20)

(17)

1

1

eF ˆ 3 Ê E ˆ Ê 2m t ∫ - Á z ,GaN z+ z ˜. Á ˜ 2 Ë Ë ¯ eF ¯

(21)

Here Ai(t ) and Bi(t ) are two independent Airy functions, and t L and t R are obtained by substituting - d / 2 and d/ 2 for z in Eq. (21), respectively. The constant C can be determined by the normalization condition. The energy eigenvalue E z of the electron confined in our wurtzite GaN/AlN QW can be obtained by solving the following equation, [ k R Ai (t R ) - KAi ¢ (t R )] [ k L Bi (t L ) + KBi ¢ (t L )]

(22)

- [ k L Ai (t L ) + KAi ¢(t L )] [ k R Bi(t R ) - KBi ¢(t R )] = 0 .

The total polaron ground-state energy can be obtained by computing the expectation value ·Y | H |Y Ò , we obtain E ( K ^ ) = ·Y | H |Y Ò =

2 K ^2 + Ez + Â Â 2 m^ j q^

[ f j ( q^ ) + f j * (q^ )]

•

Ú

dz |h( z ) |2 G j (q^ , z )

-•

q K ^ ◊ q^ ˘ È 2 + Â Â Í w j ( q ^ ) + ˙ | f j ( q^ ) | m m 2 Î ˚ j q^ ^ ^ 2

+Â

Â

j , j ¢ q^ ,q^¢

2 ^

2

2 q^ ◊ q^¢ | f j ( q^ ) |2 ◊ | f j ¢ ( q^¢ ) |2 , 2 m^

(23)

where Ez is the energy of the bare electron in z direction and can be determined by solving Eq. (22) numerically, and m ^ is the average electron effective mass and defined by •

1 1 ∫ Ú dz |h( z )|2 . m^ -• m^ ( z )

(24)

Based on variational principle, we have d E(K ^ ) d E(K ^ ) = =0. d f j ( q^ ) d f j * ( q^ )

If substituting Eq. (23) into Eq. (25), we can obtain © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

(25)

Original Paper

phys. stat. sol. (b) 242, No. 5 (2005) / www.pss-b.com •

Ú dz |h(z)| G 2

j

-•

2 q^2 2 K ^ ◊ q^ ˘ È ◊ f j* ( q^ ) (q^ , z ) + Í w j (q^ ) + m^ ˙˚ 2 m^ Î

2 q^ ◊ q^¢ | f j ¢ ( q^¢ )|2 = 0 . m^

+ f j* ( q^ ) Â Â j¢

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q^¢

(26)

For convenience, we introduce h defined as h K ^ = Â Â q^ | f j ( q^ )|2 . j

(27)

q^

Combining Eqs. (26) and (27), we have •

Ú dz |h(z)|

2

f j *( q^ ) = -

-•

w j ( q ^ ) +

G j (q^ , z )

2 q^2 2 K ^ ◊ q^ (1 - h ) 2 m^ m^

,

(28)

and È• ˘ q^ Í Ú dz |h( z )|2 G j (q^ , z ) ˙ ÎÍ -• ˚˙

h K^ = Â Â

2

. (29) 2 2 q^2 2 K ^ ◊ q^ È ˘ (1 - h ) ˙ Í w j (q^ ) + 2 m m^ Î ˚ ^ Moreover, if expanding Eq. (29) for K ^ to the first order and transforming the summation over q^ into integral, we can obtain after finishing the integral over the angle, j

q^

2

h A = 1 - h 2p

•

Â Ú dq

È• ˘ 2 3 2 Í Ú dz |h( z )| G j (q^ , z ) ˙ ◊ q^ ÎÍ -• ˚˙

. 3 2 q^2 ˘ È m ^ Í w j ( q ^ ) + 2 m^ ˙˚ Î Substituting Eqs. (26), (28) and (29) into Eq. (23), we can rewrite E ( K ^ ) as j

^

(30)

0

2

E(K ^ ) =

2 K ^2 (1 - h 2 ) + Ez - Â Â 2 m^ j q^

È• ˘ 2 Í Ú dz |h( z )| G j (q^ , z ) ˙ ÍÎ -• ˙˚ . 2 2 2 q^ K ^ ◊ q^ (1 - h ) w j (q^ ) + 2 m^ m^

(31)

Equation (31) can be calculated numerically if h is known. After transforming the last term of Eq. (31) into integral, expanding it for K ^ to the second order, and finishing the integral over angle f , we finally obtain using Eq. (30), E(K ^ ) =

2 K ^2 (1 - h ) + Ez - DE , 2 m^

(32)

where DE is the total polaron energy shift due to electron-IOP interactions and given by 2

È• ˘ q^ Í Ú dz |h( z )|2 G j (q^ , z ) ˙ • A Í -• ˙˚ DE = Â DE j = Â . dq^ Î 2 2 Ú q^ j j 2p 0 w j ( q ^ ) + 2 m^

(33)


1016


The polaron effective mass can be obtained from Eq. (32) as m^ m*^ = . 1 -h

(34)

Substituting Eq. (30) into Eq. (34), we have m*^ = m^ + Dm^ ,

(35)

where Dm^ is the total polaron mass correction due to the electron-IOP interactions, and is given by 2

Dm^ = Â Dm^, j j

3

•

A =Â dq ^ p Ú0 2 j

È• ˘ 2 3 2 Í Ú dz |h( z )| G j (q^ , z ) ˙ ◊ q^ ÎÍ -• ˚˙ 2 q^2 ˘ È Í w j (q^ ) + 2m ˙ Î ^ ˚

3

.

(36)

Numerical results and discussion

Based on our theory given in Section 2, we have performed numerical calculations for the ground-state polaron energy shift and effective mass in wurtzite GaN/AlN single QW’s including the BEF effects. The physical parameters used in our calculations are listed in Table 1. In the case of GaN and AlN, the anisotropy effect on e ( • ) and e (0) is weak, and we will assume that e ^( • ) = e z( • ) (Refs. [13– 15, 41]) and e ^(0) = e z(0) [41]. Considering the different zone-center phonon frequencies (please refer to Table 1), the anisotropy effect due to different lattice dielectric functions e ^ and e z (see Eqs. (2.6) – (2.8) of Ref. [10]) is included definitely. Moreover, we take the intensity of the BEF in the GaN/AlN single QW’s as F = 9.2 MV/cm [23]. The ratio of the conduction band offset to the band gap difference between GaN and AlN is DEc /DEg = 0.75 [42]. In order to know clearly the influence of the BEF on the polaron characteristics in wurtzite GaN/AlN QW’s, we have also calculated the polaron energy shift and the effective mass without the BEF. Figure 1 shows the polaron energy shift as a function of the QW width d. We can see from Fig. 1 that DE1 Æ 0 if d Æ 0 , DE1 is approximately a constant (4 meV) when d is large enough; DE1¢ Æ 0 if d Æ 0 or d Æ • . We know that the IOP modes disappear if d Æ 0 . Hence the polaron energy shifts DE1 and DE1¢ due to the electron interaction with the first IOP mode become zero naturally. If the BEF is neglected, |h( z )|2 has exact symmetry with respect to the center ( z = 0) of the QW and has a maximum at z = 0 . It becomes much extended if d is increased (see the dashed lines of Fig. 2). The probability of the electron appearing at the center and the two interfaces, in which G 1 (q^ , z ) has extreme values, decreases if d is increased (please refer to Figs. 2 and 3). Therefore, DE1¢ is decreased with increasing of d ( d > 4 nm). On the contrary, the maximum of |h( z )|2 will be pushed to the right interface due to the Table 1 Zone-center energies (in meV) of polar optical-phonons, optical and static dielectric constants, energy gap Eg (in eV), and electron effective mass of wurtzite AlN and GaN.

material

A1 (TO) E1 (TO) A1 (LO) a

AlN 75.72 GaN 65.91a (unstrained) GaN 67.89a (strained) a

a

83.13 69.25a

110.30 90.97a

73.22a

92.08a

Ref. [10]; bRef. [40]; cRef. [23]


a

E1 (LO) a

113.02 91.83a 94.06a

e•

e0 a

4.77 5.35a

Eg a

8.5 9.2a

mz b

6.25 0.32 3.510b 0.2b 3.645c

m^ b

0.30b 0.2b

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Fig. 1 (a) Polaron energy shift as a function of the QW width d in the case of including the BEF. The lines marked by DE , DE1 , DE2 , DE3 and DE4 indicate the total polaron energy shift, the contribution of the four different IOP branches 1, 2, 3 and 4, respectively. (b) Same as in Fig. 1(a), but for the case of ignoring BEF. The solid lines marked by DE ¢ , DE1¢ and DE4¢ indicate the total polaron energy shift and the contribution of the first and fourth IOP branches. The polaron energy shifts DE2¢ and DE3¢ arising from the second and third antisymmetric IOP branches are zero and are not shown in (b). The dotted line in (b) marked by DE corresponds to the total polaron energy shift with the BEF.

strong BEF (see the solid lines of Fig. 2). The electron is thus localized inside of the QW, close to the right interface, in which G 1 (q^ , z ) has an extreme value (see Fig. 3). Therefore, DE1 is approximately a constant if d is large enough. Furthermore, we can see from Fig. 1 that the magnitude of DE1 is obviously smaller than that of DE1¢ in the region of d < 15 nm. The physical reason is as follows. We know from our numerical calculations that G 1 (q^ , z ) has a close relation with q^ for a fixing well with d. When q^ is small (q^ < 0.4 nm -1 ), |G 1 (q^ , z )| has a maximum at the center ( z = 0) of the QW and two extrema at the interfaces (please refer to Fig. 3). If q^ is increased, its maximum will appear at the interfaces gradually. In order to explain our numerical results for DE1 and DE1¢ , we may divide the integral over q^ in Eq. (33) into two parts I1 and I 2 : I1 for the region of 0 < q^ £ q ^ and I 2 for the another region of q^ ≥ q ^ (q ^ is a definite and finite value). Thus, DE1¢ comes mainly from the contribution of I1 in the case of ignoring the BEF. Otherwise, |h( z )| 2 becomes asymmetric and its maximum is shifted to the right interface due to the BEF. So the integral I 2 dominates DE1 . Moreover, we know from Eq. (33) that the integrand decreases if q^ is increased. Hence we can understand that DE1 is smaller in magnitude than DE1¢ in the region of d £ 15 nm. Figure 1 also indicates that both DE2¢ and DE3¢ are zero for all the values of d; both DE2 and DE3 have considerable values. This is completely due to the BEF. In the case of neglecting the BEF, |h( z )| 2 is symmetric with respect to the center of the QW at z = 0. Both G 2 (q^ , z ) and G 3 (q^ , z ) are antisymmet-

Fig. 2 Probability density |h( z )|2 of the electron in the ground state as a function of the coordinate z. The solid (dashed) lines correspond to the probability density in the case of considering (neglecting) the BEF (which is supposed along the –z direction). The QW widths are taken to be d = 0.5 , 4.0 and 10.0 nm, respectively. The heterointerfaces are indicated by the vertical dotted lines.


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Yao-hui Zhu and Jun-jie Shi: Polaron effects due to interface optical-phonons Fig. 3 Electron–phonon coupling function multiplied by A as a function of the coordinate z. The lines marked by ‘Branch 1’, ‘Branch 2’, ‘Branch 3’ and ‘Branch 4’ indicate the four different IOP branches, respectively. The QW width is taken to be d = 4 nm. Here we let q^ = 0.4 nm -1. The two interfaces are located at z = -2 and 2 nm.

ric to z = 0 (please refer to Fig. 3). Hence both DE2¢ and DE3¢ are zero. On the contrary, the electron wave function is pushed to the right interface due to the strong BEF (see Fig. 2), in which both G 2 (q^ , z ) and G 3 (q^ , z ) have maximum values. Hence DE2 and DE3 have considerable values in the case of including the BEF. Compared with the case of the first branch, the contribution of the fourth branch to the polaron energy shift has a similar dependent relation with the well width d. This is because both the first and the fourth IOP modes are symmetric modes with respect to the center z = 0 of the QW. The difference of magnitudes between the first and the fourth IOP modes can be completely understood based on the following fact. The fourth mode is always symmetric interface mode which has its maxima at the two interfaces of the QW. Otherwise, the first mode has its maximum at the center z = 0 of the QW if q^ is small (q^ < 0.4 nm -1 ) and becomes interface modes which has its maxima at the two interfaces of the QW when q^ is large. Moreover, Fig. 1 also clearly shows that the BEF has a small influence on the total polaron energy shift in the case of the small well width d ( d < 6 nm). Otherwise, the BEF has a remarkable influence on the polaron effects if d is large enough. Considering the effect of the BEF due to the spontaneous and the piezoelectric polarizations of GaN/AlN QW’s, the total polaron energy shift is increased slowly and basically approaches to a considerable constant. On the contrary, it will monotonically decrease to zero if d is large enough in the case of ignoring the BEF. The physical reason is that |h( z )| 2 is pushed from the center to the right interface of the QW by the strong BEF with increasing the well width d (see Fig. 2). The electron-IOP interactions will thus be enhanced. The calculation results for the polaron effective mass m*^ and average electron effective mass m^ as a function of the GaN/AlN QW width d are shown in Fig. 3. We note from Fig. 3 that the bulk AlN electron effective mass m^,AlN = 0.3 is exactly obtained if d Æ 0 . This is because the system is reduced to the AlN bulk material and the polaron effects due to the electron-IOP interactions disappear in the limiting case d Æ 0 . Moreover, the electron will be completely confined in the GaN well layer when d is large enough without the BEF. Hence m^ will approach to m^,GaN = 0.2 if d Æ • . On the contrary, the electron will be strongly localized in the vicinity of the right interface of the QW due to the BEF and keep a considerable probability to appear in the AlN barrier layer. This will directly lead to the average electron effective mass becoming a larger constant than the situation without the BEF if d is large enough. Figure 4 also indicates that the polaron effective mass approaches to the GaN electron effective mass 0.2 if the BEF is ignored and the well width d is large enough. Considering the BEF, the electron is mainly localized in the vicinity of the GaN/AlN heterointerface and the polaron effective mass thus becomes a larger constant of 0.233 if d > 10 nm. This value is reasonable compared with the experimental measurements of Refs. [24– 26]. The difference between the two cases is completely due to the BEF effects. The physical reason is similar to the situation of the polaron energy shift due to the electron-IOP interactions and has been deeply analyzed in Fig. 1. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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1019 Fig. 4 Polaron effective mass as a function of the QW width d. The solid (dotted) line marked by m*⊥ shows the polaron effective mass with (without) the BEF. The solid (dotted) line marked by m^ corresponds to the average electron effective mass in the case of considering (neglecting) the BEF. Here m0 is free electron mass.

It is more interesting to compare the current results for GaN/AlN QW’s with the corresponding results for the GaAs/AlGaAs QW’s [43 – 45]. Generally, compared with the cubic Q2D systems, such as GaAs/AlGaAs, the optical-phonon modes in a wurtzite Q2D multilayer heterostructure are much more complicated due to the anisotropy of wurtzite crystals. It is well known that, in cubic Q2D multilayer heterostructures, there are only three types of the optical-phonon modes, i.e., the interface modes, the confined modes, and the half-space modes [43– 45]. The results of Refs. [10 –12] clearly indicate that two new phonon modes, i.e., the propagating modes and the quasiconfined modes, together with the interface modes, the exactly confined modes, and the half-space modes, coexit in wurtzite Q2D multilayer systems due to the anisotropy of wurtzite crystals. Hence the polaron effects in a wurtzite QW become much more complicated than those in a cubic QW. To the best of our knowledge, understanding of the phonon modes and their interactions with electrons in wurtzite Q2D quantum heterostructures is still incomplete up to now. We will thus only pay our attention to the polaron effects due to the IOP’s in GaN/AlN quantum wells in the present article. It is worthwhile to note that the polaron effects due to electron-IOP interactions are significant because the wave functions of the electron confined in GaNbased QW’s will be pushed close to the one interface due to the strong BEF [9] and the electron is thus mainly localized at the one interface of the QW’s, which is quite different from the case in GaAs/AlGaAs QW’s. The contribution of the IOP’s to the polaron effects always important for GaN/AlN QW’s with the well width from d = 0.5 nm to d Æ • (please refer to Figs. 1 and 2). Moreover, we find from Figs. 1 and 4 that the polaron effects in GaN-based QW’s are extremely strong. The polaron energy shift due to the electron-IOP interactions in GaN/AlN QW’s is approximately 60 meV, which is much larger than that in GaAs/AlGaAs QW’s (approximately 3 meV). Hence the polaron effect is an important effect in GaN-based quantum heterostructures and should be considered seriously. We will investigate the polaron effects due to the other phonon modes besides the IOP modes in GaN-based QW’s if the corresponding phonon modes are solved in the future. We believe that the polaron correction for the bulk AlN (GaN) can be obtained in the limiting case of the well width d Æ 0 (d Æ •) if all of the phonon modes are included and the BEF effect is ignored. By the way, no experimental results are available for the polaron effects in the GaN-based QW’s at present. We expect that the current polaron correction due to IOP’s in GaN-based QW’s can be confirmed by using a cyclotron resonant technique as suggested by Ref. [46]. It has been known that the single polaron theory is expected to give a satisfactory description to some experimental results obtained from the samples with low electron density, such as ne £ 1011 cm -2 in GaAs/AlGaAs QW’s [46]. However, the many-particle effects become significant and should be considered if the electron density is high. Generally, the polaron effects can be reduced due to screening in a sample with high electron density [27].

4

Conclusions

In this paper, we investigated the polaron energy shift and the effective mass due to the electron-IOP interactions in wurtzite GaN/AlN single QW’s by using the LLP variational approach. The BEF effects © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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due to the spontaneous and the piezoelectric polarizations of QW’s, the anisotropic effects of wurtzite crystals and the strains of the QW’s are included. Our numerical results show that the strong BEF has a remarkable influence on the polaron effects for GaN/AlN QW’s with a large well width d. The polaron energy shift increases slowly and the polaron effective mass approaches to a constant if the well width d is large enough. On the contrary, both the polaron energy shift and the effective mass decrease slowly with the increasing of d ( d > 4 nm), and the polaron energy shift approaches zero when d is large enough without the BEF effects. The physical reason has been analyzed in depth. The polaron effects in GaNbased quantum heterostructures are quite strong and need to be investigated seriously in the future. The results obtained in the present paper are very useful for further investigating the optical and transport properties of the GaN-based QW’s and have significant practical meanings for some important optoelectric devices, such as LED’s and LD’s, designs and applications. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant Nos. 60276004 and 60390073, and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China.

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